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I N T R O D U C T I O N

The function concept is one of the most important ideas in mathematics. The study

of mathematics beyond the elementary level requires a firm understanding of a basic

list of elementary functions, their properties, and their graphs. See the inside front

cover of this book for a list of the functions that form our library of elementary func-

tions. Most functions in the list will be introduced to you by the end of Chapter 2 and

should become a part of your mathematical toolbox for use in this and most future

courses or activities that involve mathematics. A few more elementary functions

may be added to these in other courses, but the functions listed inside the front cover

are more than sufficient for all the applications in this text.

2-1 Functions

2-2 Elementary Functions:Graphs andTransformations

2-3 Quadratic Functions

2-4 Exponential Functions

2-5 Logarithmic Functions

Chapter 2 Review

Review Exercise

2CHAPTER

Functions and Graphs

BARNMC02_0132255707.QXD 12/8/06 5:10 PM Page 46

S e c t i o n 2 . 1 Functions 47

FUNCTIONS� Equations in Two Variables� Definition of a Function� Functions Specified by Equations� Function Notation� Applications

We introduce the general notion of a function as a correspondence between two sets.Then we restrict attention to functions for which the two sets are both sets of real num-bers. The most useful are those functions that are specified by equations in two vari-ables. We discuss the terminology and notation associated with functions, graphs offunctions, and applications to economics.

� Equations in Two VariablesIn Chapter 1 we found that the graph of an equation of the form where A and B are not both zero, is a line. Because a line is determined by any twoof its points, such an equation is easy to graph: Just plot any two points in its solutionset and sketch the unique line through them.

More complicated equations in two variables, for example, or are more difficult to graph.To sketch the graph of an equation, we plot enough pointsfrom its solution set in a rectangular coordinate system so that the total graph is ap-parent and then connect these points with a smooth curve.This process is called point-by-point plotting.

x2= y4,y = 9 - x2

Ax + By = C,

Section 2-1

Point-by-Point Plotting Sketch the graph of each equation.

(A) (B) x2= y4y = 9 - x2

(A) Make up a table of solutions—that is, ordered pairs of real numbers that satisfythe given equation. For easy mental calculation, choose integer values for x.

SOLUTIONS

E X A M P L E 1

5�5

�5

5

10

�10

�10 10(3, 0)

(2, 5)

(�3, 0)

(�2, 5)

(4, �7)(�4, �7)

(0, 9)(1, 8)(�1, 8)

y � 9 � x2

y

x

FIGURE 1 y = 9 - x2

x 0 1 2 3 4

y 0 5 8 9 8 5 0 -7-7

-1-2-3-4

After plotting these solutions, if there are any portions of the graph that are un-clear, plot additional points until the shape of the graph is apparent. Then joinall the plotted points with a smooth curve as shown in Figure 1.Arrowheads are


To graph the equation , we use point-by-point plotting to obtainy = -x3+ 3x

48 C H A P T E R 2 Functions and Graphs

FIGURE 2 x2= y4

5�5

�5

5

10

�10

�10 10

y

x

x2 � y4

5

�5

�5 5x

y

Explore & Discuss 1

(A) Do you think this is the correct graph of the equation? If so, why? If not,why?

(B) Add points on the graph for

(C) Now, what do you think the graph looks like? Sketch your version of thegraph, adding more points as necessary.

(D) Graph this equation on a graphing calculator and compare it with yourgraph from part (C).

The icon in the margin is used throughout this book to identify optional graphingcalculator activities that are intended to give you additional insight into the conceptsunder discussion. You may have to consult the manual for your graphing calculatorfor the details necessary to carry out these activities. For example, to graph the equa-tion in Explore–Discuss 1 on most graphing calculators, you first have to enter theequation (Fig. 3A) and the window variables (Fig. 3B).

As Explore–Discuss 1 illustrates, the shape of a graph may not be apparent fromyour first choice of points on the graph. Using point-by-point plotting, it may bedifficult to find points in the solution set of the equation, and it may be difficult todetermine when you have found enough points to understand the shape of the graph.We will supplement the technique of point-by-point plotting with a detailed analysis

x = -2, -1.5, -0.5, 0.5, 1.5, and 2.

FIGURE 3

(A)

(B)

x y

�1 �2

0 0

1 2

used to indicate that the graph continues beyond the portion shown here withno significant changes in shape.

(B) Again we make a table of solutions—here it may be easier to choose integervalues for y and calculate values for x. Note, for example, that if then

that is, the ordered pairs and are both in the solutionset.

(-4, 2)(4, 2)x = ;4;y = 2,

x 0

y 0 1 2 3-1-2-3

;9;4;1;1;4;9

We plot these points and join them with a smooth curve (Fig. 2).

MATCHED PROBLEM 1 Sketch the graph of each equation.

(A) (B) y2=

100

x2+ 1

y = x2- 4



of several basic equations, giving you the ability to sketch graphs with accuracy andconfidence.

� Definition of a FunctionCentral to the concept of function is correspondence. You have already had experi-ences with correspondences in daily living. For example,

To each person there corresponds an annual income.

To each item in a supermarket there corresponds a price.

To each student there corresponds a grade-point average.

To each day there corresponds a maximum temperature.

For the manufacture of x items there corresponds a cost.

For the sale of x items there corresponds a revenue.

To each square there corresponds an area.

To each number there corresponds its cube.

One of the most important aspects of any science is the establishment of correspon-dences among various types of phenomena. Once a correspondence is known, pre-dictions can be made. A cost analyst would like to predict costs for various levels ofoutput in a manufacturing process; a medical researcher would like to know the cor-respondence between heart disease and obesity; a psychologist would like to predictthe level of performance after a subject has repeated a task a given number of times;and so on.

What do all the examples above have in common? Each describes the matchingof elements from one set with the elements in a second set. Consider the tables of thecube, square, and square root given in Tables 1–3.

Tables 1 and 2 specify functions, but Table 3 does not. Why not? The definition ofthe term function will explain.

Number Cube

TABLE 1

RangeDomain

0 01 12 8

�1�1�8�2

Number Square

TABLE 2

RangeDomain

012

�1�2

Number Square root

TABLE 3

RangeDomain

4

10

0 01

1

24

39

�1

�2

�3

FunctionA function is a correspondence between two sets of elements such that to eachelement in the first set there corresponds one and only one element in the secondset.

The first set is called the domain, and the set of corresponding elements in thesecond set is called the range.

DEFINITION

Tables 1 and 2 specify functions, since to each domain value there corresponds ex-actly one range value (for example, the cube of is and no other number). Onthe other hand,Table 3 does not specify a function, since to at least one domain valuethere corresponds more than one range value (for example, to the domain value 9there corresponds and 3, both square roots of 9).-3

-8-2

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Consider the set of students enrolled in a college and the set of faculty membersof that college. Suppose we define a correspondence between the two sets bysaying that a student corresponds to a faculty member if the student is currentlyenrolled in a course taught by that faculty member. Is this correspondence a func-tion? Discuss.

� Functions Specified by EquationsMost of the functions in this book will have domains and ranges that are (infinite) setsof real numbers. The graph of such a function is the set of all points in theCartesian plane such that x is an element of the domain and y is the correspondingelement in the range. The correspondence between domain and range elements isoften specified by an equation in two variables. Consider, for example, the equationfor the area of a rectangle with width 1 inch less than its length (Fig. 4). If x is thelength, then the area y is given by

For each input x (length), we obtain an output y (area). For example,

If then

If then

If then

.

The input values are domain values, and the output values are range values. Theequation assigns each domain value x a range value y. The variable x is called anindependent variable (since values can be “independently” assigned to x from the do-main), and y is called a dependent variable (since the value of y “depends” on thevalue assigned to x). In general, any variable used as a placeholder for domain val-ues is called an independent variable; any variable that is used as a placeholder forrange values is called a dependent variable.

When does an equation specify a function?

L 2.7639

y = 15 (15 - 1) = 5 - 15x = 15,

y = 1 (1 - 1) = 1 # 0 = 0.x = 1,

y = 5 (5 - 1) = 5 # 4 = 20.x = 5,

y = x(x - 1) x Ú 1

(x, y)

Explore & Discuss 2

x � 1

x

FIGURE 4

Functions Specified by EquationsIf in an equation in two variables, we get exactly one output (value for the depen-dent variable) for each input (value for the independent variable), then the equa-tion specifies a function. The graph of such a function is just the graph of thespecifying equation.

If we get more than one output for a given input, the equation does not specify afunction.

DEFINITION

Functions and Equations Determine which of the following equations specifyfunctions with independent variable x.

(A) x a real number (B) x a real numbery2- x2

= 9,4y - 3x = 8,

E X A M P L E 2

(A) Solving for the dependent variable y, we have

(1)

y = 2 +

34

x

4y = 8 + 3x

4y - 3x = 8

SOLUTION

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* Recall that each positive real number N has two square roots: the principal square root, and

the negative of the principal square root (see Appendix A, Section A-6).-2N,

2N,

Since each input value x corresponds to exactly one output value we see that equation (1) specifies a function.

(B) Solving for the dependent variable y, we have

(2)

Since is always a positive real number for any real number x and sinceeach positive real number has two square roots,* to each input value x there corresponds two output values ( and For ex-ample, if then equation (2) is satisfied for and for Thus,equation (2) does not specify a function.

y = -5.y = 5x = 4,y = 29 + x2).y = -29 + x2

9 + x2

y = ;29 + x2

y2= 9 + x2

y2- x2

= 9

(y = 2 +34x),

Since the graph of an equation is the graph of all the ordered pairs that satisfy theequation, it is very easy to determine whether an equation specifies a function by ex-amining its graph. The graphs of the two equations we considered in Example 2 areshown in Figure 5.

x5�5

�5

5

10

�10

�10 10

y

x5�5

�5

5

10

�10

�10 10

y

(A) 4y � 3x � 8 (B) y2 � x2 � 9

FIGURE 5

In Figure 5A notice that any vertical line will intersect the graph of the equationin exactly one point. This shows that to each x value there corresponds

exactly one y value and confirms our conclusion that this equation specifies a function.On the other hand, Figure 5B shows that there exist vertical lines that intersect thegraph of in two points. This indicates that there exist x values to whichthere correspond two different y values and verifies our conclusion that this equationdoes not specify a function. These observations are generalized in Theorem 1.

y2- x2

= 9

4y - 3x = 8

THEOREM 1 VERTICAL-LINE TEST FOR A FUNCTIONAn equation specifies a function if each vertical line in the coordinate system passesthrough at most one point on the graph of the equation.

If any vertical line passes through two or more points on the graph of an equation,then the equation does not specify a function.

MATCHED PROBLEM 2 Determine which of the following equations specify functions with independentvariable x.

(A) x a real number (B) x a real number3y - 2x = 3,y2- x4

= 9,



The function graphed in Figure 5A is an example of a linear function.The vertical-line test implies that equations of the form , where , specify func-tions; they are called linear functions. Similarly, equations of the form specifyfunctions; they are called constant functions, and their graphs are horizontal lines.The vertical-line test implies that equations of the form do not specify functions;note that the graph of is itself a vertical line.x = a

x = a

y = bm Z 0y = mx + b

Explore & Discuss 3 The definition of a function specifies that to each element in the domain there cor-responds one and only one element in the range.

(A) Give an example of a function such that to each element of the range therecorrespond exactly two elements of the domain.

(B) Give an example of a function such that to each element of the rangethere corresponds exactly one element of the domain.

In Example 2, the domains were explicitly stated along with the given equations.In many cases, this will not be done. Unless stated to the contrary, we shall adhere tothe following convention regarding domains and ranges for functions specified byequations:

If a function is specified by an equation and the domain is not indicated, then weassume that the domain is the set of all real number replacements of the independentvariable (inputs) that produce real values for the dependent variable (outputs). Therange is the set of all outputs corresponding to input values.

In many applied problems the domain is determined by practical considerationswithin the problem (see Example 7).

Finding a Domain Find the domain of the function specified by the equationassuming that x is the independent variable.y = 14 - x,

For y to be real, must be greater than or equal to 0; that is,

Sense of inequality reverses when both sides are divided by �1.

Thus,

Domain: (inequality notation) or (interval notation)(-q, 4]x … 4

x … 4

- x Ú -4

4 - x Ú 0

4 - xSOLUTION

E X A M P L E 3

MATCHED PROBLEM 3 Find the domain of the function specified by the equation , assumingx is the independent variable.

� Function NotationWe have just seen that a function involves two sets, a domain and a range, and a cor-respondence that assigns to each element in the domain exactly one element in therange. We use different letters to denote names for numbers; in essentially the sameway, we will now use different letters to denote names for functions. For example,f and g may be used to name the functions specified by the equations

and

(3) g: y = x2

+ 2x - 3

f: y = 2x + 1

y = x2+ 2x - 3:

y = 2x + 1

y = 2x - 2



If x represents an element in the domain of a function f, then we frequently usethe symbol

in place of y to designate the number in the range of the function f to which x is paired(Fig. 6). This symbol does not represent the product of f and x. The symbol f(x) isread as “f of x,” “f at x,” or “the value of f at x.” Whenever we write , weassume that the variable x is an independent variable and that both y and f(x) aredependent variables.

Using function notation, we can now write functions f and g in (3) in the form

Let us find f(3) and . To find f(3), we replace x with 3 wherever x occurs inand evaluate the right side:

For input 3, the output is 7.

Thus,

The function f assigns the range value 7 to the domain value 3.

To find , we replace each x by in and evaluate theright side:

For input �5, the output is 12.

Thus,

The function g assigns the range value 12 to the domain value �5.

It is very important to understand and remember the definition of f(x):

For any element x in the domain of the function f, the symbol f(x) represents theelement in the range of f corresponding to x in the domain of f. If x is an input value,then f(x) is the corresponding output value. If x is an element that is not in the domainof f, then f is not defined at x and f(x) does not exist.

g(-5) = 12

= 25 - 10 - 3 = 12

g(-5) = (-5)2+ 2(-5) - 3

g(x) = x2+ 2x - 3

g(x) = x2+ 2x - 3-5g(-5)

f(3) = 7

= 6 + 1 = 7

f(3) = 2 # 3 + 1

f(x) = 2x + 1

f(x) = 2x + 1g(-5)

f(x) = 2x + 1 and g(x) = x2+ 2x - 3

y = f(x)

f (x)

f

x f (x)

DOMAIN RANGE

FIGURE 6

Function Evaluation If

then

(A)*

(B)

(C)

But is not a real number. Since we have agreed to restrict the domain of afunction to values of x that produce real values for the function, is not in thedomain of h and does not exist.

(D)

= -6 - 3 = -9

=

12-2

+ 0 - 29

=

120 - 2

+ (1 - 12) - 210 - 1f(0) + g(1) - h(10)

h(-2)-2

1-3

= 1-3= 1-2 - 1h(-2)

= 1 - 4 = -3= 1 - (-2)2g(-2)

=

124

= 3=

126 - 2

f(6)

f(x) =

12x - 2

g(x) = 1 - x2 h(x) = 2x - 1

E X A M P L E 4

* Dashed boxes are used throughout the book to represent steps that are usually performed mentally.



MATCHED PROBLEM 4 Use the functions in Example 3 to find

(A) (B) (C) (D)f(3)

h(5)h(-8)g(-1)f(-2)

Finding Domains Find the domains of functions f, g, and h:

f(x) =

12x - 2

g(x) = 1 - x2 h(x) = 2x - 1

Domain of f: represents a real number for all replacements of x by realnumbers except for (division by 0 is not defined).Thus, f(2) does not exist, andthe domain of f is the set of all real numbers except 2.We often indicate this by writing

Domain of g: The domain is R, the set of all real numbers, since represents areal number for all replacements of x by real numbers.

Domain of h: The domain is the set of all real numbers x such that is a realnumber—that is, such that

x Ú 1 or 31, q )

x - 1 Ú 0

2x - 1

1 - x2

f(x) =

12x - 2

x Z 2

x = 212>(x - 2)SOLUTION

E X A M P L E 5

MATCHED PROBLEM 5 Find the domains of functions F, G, and H:

In addition to evaluating functions at specific numbers, it is important to be ableto evaluate functions at expressions that involve one or more variables. For example,the difference quotient

x and x � h in the domain of f,

is studied extensively in calculus.

h Z 0f(x + h) - f(x)

h

F(x) = x2- 3x + 1 G(x) =

5x + 3

H(x) = 22 - x

In algebra, you learned to use parentheses for grouping variables. For example,

Now we are using parentheses in the function symbol . For example, if then

Note that . That is, the function name f does not dis-tribute across the grouped variables as the “2” does in (see AppendixA, Section A-2).

2(x + h)(x + h)x2

+ h2Z f(x + h)f(x) + f(h) =

= x2+ 2xh + h2f(x + h) = (x + h)2

f(x) = x2,f(x)

2(x + h) = 2x + 2h

I N S I G H T

Explore & Discuss 4 Let x and h be real numbers.

(A) If which of the following is true?1.2.3. f(x + h) = 4x + 4h + 6

f(x + h) = 4x + 4h + 3f(x + h) = 4x + 3 + h

f(x) = 4x + 3,



APPLICATIONSWe now turn to the important concepts of break-even and profit–loss analysis, whichwe will return to a number of times in this book. Any manufacturing company hascosts, C, and revenues, R. The company will have a loss if will break even if

, and will have a profit if Costs include fixed costs such as plant over-head, product design, setup, and promotion; and variable costs, which are dependenton the number of items produced at a certain cost per item. In addition, price–demandfunctions, usually established by financial departments using historical data or sam-pling techniques, play an important part in profit–loss analysis.We will let x, the num-ber of units manufactured and sold, represent the independent variable. Costfunctions, revenue functions, profit functions, and price–demand functions are oftenstated in the following forms, where a, b, m, and n are constants determined from thecontext of a particular problem:

Cost Function

Price–Demand Function

x is the number of items that can be sold at $p per item.

Revenue Function

Profit Function

= x(m - nx) - (a + bx)P = R - C

= xp = x(m - nx)R = (number of items sold) * (price per item)

p = m - nx

= a + bx

C = (fixed costs) + (variable costs)

R 7 C.R = CR 6 C,

Using Function Notation find

(A) (B) (C) (D)f(a + h) - f(a)

h, h Z 0f(a + h) - f(a)f(a + h)f(a)

For f(x) = x2- 2x + 7,E X A M P L E 6

(A)(B)

(C)

(D)

= 2a + h - 2

Because h Z 0, hh

= 1.h(2a + h - 2)

h

f(a + h) - f(a)

h=

2ah + h2- 2h

h=

= 2ah + h2- 2h

f(a + h) - f(a) = (a2+ 2ah + h2

- 2a - 2h + 7) - (a2- 2a + 7)

f(a + h) = (a + h)2- 2(a + h) + 7 = a2

+ 2ah + h2- 2a - 2h + 7

f(a) = a2- 2a + 7SOLUTION

MATCHED PROBLEM 6 Repeat Example 6 for f(x) = x2- 4x + 9.

(B) If , which of the following is true?

1.2.3.

(C) If describe the operations that must be performed toevaluate .M(x + h)

M(x) = x2+ 4x + 3,

g(x + h) = x2+ 2hx + h2

g(x + h) = x2+ h2

g(x + h) = x2+ h

g(x) = x2



Example 7 and Matched Problem 7 explore the relationships among the algebraicdefinition of a function, the numerical values of the function, and the graphical rep-resentation of the function.The interplay among algebraic, numeric, and graphic view-points is an important aspect of our treatment of functions and their use. In Example 7,we also see how a function can be used to describe data from the real world, a processthat is often referred to as mathematical modeling. The material in this example willbe returned to in subsequent sections so that we can analyze it in greater detail andfrom different points of view.

Price–Demand and Revenue Modeling A manufacturer of a popular digitalcamera wholesales the camera to retail outlets throughout the United States. Usingstatistical methods, the financial department in the company produced the price–demand data in Table 4, where p is the wholesale price per camera at which x millioncameras are sold. Notice that as the price goes down, the number sold goes up.

(A)SOLUTION

E X A M P L E 7

TABLE 4 Price–Demand

x (Millions) p ($)

2 87

5 68

8 53

12 37

Using special analytical techniques (regression analysis), an analyst arrived at thefollowing price–demand function that models the Table 4 data:

(5)

(A) Plot the data in Table 4.Then sketch a graph of the price–demand function in thesame coordinate system.

(B) What is the company’s revenue function for this camera, and what is the domainof this function?

(C) Complete Table 5, computing revenues to the nearest million dollars.

(D) Plot the data in Table 5.Then sketch a graph of the revenue function using thesepoints.

(E) Plot the revenue function on a graphing calculator.

p(x) = 94.8 - 5x 1 … x … 15

TABLE 5 Revenue

x (Millions) R(x) (Millions $)

1 90

3

6

9

12

15

p(x)

5 10 15

50

100

Million cameras

Pric

e pe

r ca

mer

a ($

)

x

FIGURE 7 Price–demand

In Figure 7, notice that the model approximates the actual data in Table 4, andit is assumed that it gives realistic and useful results for all other values of xbetween 1 million and 15 million.



(B) million dollars

[Same domain as the price–demand function, equation (5).]

(C)

Domain: 1 … x … 15

R(x) = xp(x) = x(94.8 - 5x)

TABLE 5 Revenue

x (Millions) R(x) (Million $)

1 90

3 239

6 389

9 448

12 418

15 297

(D) (E)

0

1

500

15

x

R(x)

50 10 15

Million cameras

100

200

300

400

500M

illio

n do

llars

MATCHED PROBLEM 7 The financial department in Example 6, using statistical techniques, produced thedata in Table 6, where C(x) is the cost in millions of dollars for manufacturing andselling x million cameras.

TABLE 6 Cost Data

x (Millions) C(x) (Million $)

1 175

5 260

8 305

12 395

TABLE 7 Profit

x (Millions) P(x) (Million $)

1 �86

3

6

9

12

15

Using special analytical techniques (regression analysis), an analyst producedthe following cost function to model the data:

(6)

(A) Plot the data in Table 6. Then sketch a graph of equation (6) in the same coor-dinate system.

(B) What is the company’s profit function for this camera, and what is its domain?

(C) Complete Table 7, computing profits to the nearest million dollars.

C(x) = 156 + 19.7x 1 … x … 15



1. (A) (B)

2. (A) Does not specify a function

(B) Specifies a function

3. (inequality notation) or (interval notation)

4. (A)

(B) 0

(C) Does not exist

(D) 6

5. Domain of F: R; domain of G: all real numbers except domain of (inequality notation) or (interval notation)

6. (A)

(B)

(C)

(D)

7. (A)

2a + h - 4

2ah + h2- 4h

a2+ 2ah + h2

- 4a - 4h + 9

a2- 4a + 9

(- q, 2]H: x … 2-3;

-3

32, q)x Ú 2

�5�10 5 10

5

15

10

�5

y � x2 � 4y

x

Answers to Matched Problems

x

C(x)

50 10 15

100

200

300

400

500

Million cameras

Mill

ion

dolla

rs

(B)

(C)

P(x) = R(x) - C(x) = x(94.8 - 5x) - (156 + 19.7x); domain: 1 … x … 15

TABLE 7 Profit


1 �86

3 24

6 115

9 115

12 25

15 �155

�5

�10

5

5

10

�5

y

x

(D) Plot the points from part (C).Then sketch a graph of the profit function throughthese points.

(E) Plot the profit function on a graphing calculator.



A In Problems 1–8, use point-by-point plotting to sketch thegraph of each equation.

1.

2.

3.

4.

5.

6.

7.

8.

Indicate whether each table in Problems 9–14 specifies a function.

9. 10.

11. 12.

13. 14.

Indicate whether each graph in Problems 15–20 specifies afunction.

15. 16.

xy = 12

xy = -6

x = y3

y = x3

y = x2

x = y2

x = y + 1

y = x + 1

17. 18.

19. 20.

In Problems 21–30, each equation specifies a function.Determine whether the function is linear, constant, or neither.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

In Problems 31 and 32 which of the indicated correspondencesdefine functions? Explain.

31. Let P be the set of residents of Pennsylvania and let R and S be the set of members of the U.S. House ofRepresentatives and the set of members of the U.S. Senate,respectively, elected by the residents of Pennsylvania.

y = x2- 9

y = x2+ (1 - x)(1 + x) + 1

y =

12x + 3

y =

1 + x

2+

1 - x

3

y = 3x +12(5 - 6x)

y = 8x - 1 +

1x

y = 5x -12(4 - x)

y = -6

y = p

y = 3 - 7x

Exercise 2-1

RangeDomain

RangeDomain

RangeDomain

3 05 17 2

3 56

4 75 8

35

6

96

12

RangeDomain

RangeDomain

RangeDomain

579

8 09 1

210 3

60

1

�3�2�1

�2

�1

x

y

�5

�5

�10

�10 5 10

5

10

x

y

�5

�5

�10

�10 5 10

5

10

x

y

�5

�5

�10

�10 5 10

5

10

x

y

�5

�5

�10

�10 5 10

5

10

x

y

�5

�5

�10

�10 5 10

5

10

x

y

�5

�5

�10

�10 5 10

5

10

(D) (E)

x

P(x)

5 100 15

100

200

�200

�100Mill

ion

dolla

rs

Million cameras

�200

1

200

15



(A) A resident corresponds to the congressperson rep-resenting the resident’s congressional district.

(B) A resident corresponds to the senator representingthe resident’s state.

32. Let P be the set of patients in a hospital, let D be the setof doctors on the hospital staff, and N be the set ofnurses on the hospital staff.

(A) A patient corresponds to the doctor if that doctoradmitted the patient to the hospital.

(B) A patient corresponds to the nurse if that nursecares for the patient.

In Problems 33–40, use point-by-point plotting to sketch thegraph of each function.

33. 34.

35. 36.

37. 38.

39. 40.

In Problems 41 and 42, the three points in the table are on thegraph of the indicated function f. Do these three points providesufficient information for you to sketch the graph of Add more points to the table until you are satisfied that yoursketch is a good representation of the graph of on theinterval [-5, 5].

y = f(x)

y = f(x)?

f(x) =

-6x

f(x) =

8x

f(x) = x3- 2f(x) = 4 - x3

f(x) = 3 - x2f(x) = x2- 1

f(x) =

x

2- 3f(x) = 1 - x

45. 46.

47. 48.

49. 50.

51. 52.

If and find each of the expres-sions in Problems 53–64.

53. f(2) 54. f(1)

55. 56. g(1)

57. 58.

59. 60.

61. 62.

63. 64.

B In Problems 65–70, find the domain of each function.

65. 66.

67. 68.

69. 70.

71. Two people are discussing the function

and one says to the other, “f(2) exists but f(3) does not.”Explain what they are talking about.

72. Referring to the function in Problem 71, do andexist? Explain.

The verbal statement “function f multiplies the square of thedomain element by 3 and then subtracts 7 from the result” andthe algebraic statement “ define the samefunction. In Problems 73–76, translate each verbal definitionof a function into an algebraic definition.

73. Function g subtracts 5 from twice the cube of the domainelement.

74. Function f multiplies the domain element by andadds 4 to the result.

75. Function G multiplies the square root of the domain ele-ment by 2 and subtracts the square of the domain ele-ment from the result.

76. Function F multiplies the cube of the domain elementby and adds 3 times the square root of 3 to theresult.

In Problems 77–80, translate each algebraic definition of thefunction into a verbal definition.

77. 78.

79. 80.

Determine which of the equations in Problems 81–90 specifyfunctions with independent variable x. For those that do, find

G(x) = 41x - x2F(x) = 3x3- 21x

g(x) = -2x + 7f(x) = 2x - 3

-8

-3

f(x) = 3x2- 7”

f(-3)f(-2)

f(x) =

x2- 4

x2- 9

F(x) =

1

25 + xg(x) = 27 - x

g(x) =

x + 1x - 2

f(x) =

x - 2x + 4

H(x) = 7 - 2x2- x4F(x) = 2x3

- x2+ 3

g(-3)

f(2)

g(-2)

f(-2)

g(0) # f(-2)g(3) # f(0)

f(3) - g(3)f(1) + g(2)

g(-2)g(-3)

f(-1)

g(x) = x2+ 2x,f(x) = 2x - 3

4 = f(x)-4 = f(x)

3 = f(x), x 6 00 = f(x)

y = f(-2)y = f(5)

y = f(4)y = f(-5)

f(x) =

2x

x2+ 1

42. x 0 1 2

f(x) 0 1 2f(x) =

3x2

x2+ 2

43. Let and

(A) Evaluate f(x), g(x), and for 5,10, 15, 20.

(B) Graph and onthe interval [0, 20].

44. Repeat Problem 43 for and

In Problems 45–52, use the following graph of a function f todetermine x or y to the nearest integer, as indicated. Some prob-lems may have more than one answer.

g(x) = 200 + 50x.f(x) = 200x - 10x2

g(x)y = f(x) -y = f(x), y = g(x),

x = 0,f(x) - g(x)

0 … x … 20.g(x) = 150 + 20x,f(x) = 100x - 5x2

x

f (x)

�5

�5

�10

�10 5 10

5

10

y � f (x)

41. x �1 0 1

f(x) �1 0 1



the domain. For those that do not, find a value of x to whichthere corresponds more than one value of y.

81. 82.

83. 84.

85. 86.

87. 88.

89. 90.

91. If find

92. If find

93. If find

94. If find

If find and simplify each expression in Prob-lems 95–106.

95. f(5) 96.

97. 98.

99. 100.

101. 102.

103. f(2x) 104.

105. 106. f(1 - x)f(x + 1)

f(-3x)

f(f(-2))f(f(1))

f(3) - f(6)f(2) + f(5)

f(3 - 6)f(2 + 5)

f(-3)

f(x) = x2- 1,

P(3 + h) - P(3)

h

P(x) = 2x2- 3x - 7,

Q(2 + h) - Q(2)

h

Q(x) = x2- 5x + 1,

G(2 + h) - G(2)

h

G(r) = 3 - 5r,

F(3 + h) - F(3)

h

F(t) = 4t + 7,

x2- y2

= 16x2+ y2

= 25

xy + y - x = 5xy - 4y = 1

x2+ y = 10x + y2

= 10

x - y2= 1x2

- y = 1

3y - 7x = 154x - 5y = 20

C In Problems 107–112, find and simplify each of the following.

(A)

(B)

(C)

107. 108.

109. 110.

111. 112.

Problems 113–116, refer to the area A and perimeter P of a rec-tangle with length l and width w (see the figure).

f(x) = x(x + 40)f(x) = x(20 - x)

f(x) = 3x2+ 5x - 8f(x) = 4x2

- 7x + 6

f(x) = -3x + 9f(x) = 4x - 3

f(x + h) - f(x)

h

f(x + h) - f(x)

f(x + h)

w

l

A � lw

P � 2l � 2w

113. The area of a rectangle is 25 square inches. Express theperimeter P(w) as a function of the width w, and statethe domain of this function.

114. The area of a rectangle is 81 square inches. Express theperimeter P(l) as a function of the length l, and state thedomain of this function.

115. The perimeter of a rectangle is 100 meters. Express thearea A(l) as a function of the length l, and state the do-main of this function.

116. The perimeter of a rectangle is 160 meters. Express thearea A(w) as a function of the width w, and state the do-main of this function.

117. Price–demand. A company manufactures memory chips formicrocomputers. Its marketing research department, using sta-tistical techniques, collected the data shown in Table 8, wherep is the wholesale price per chip at which x million chips canbe sold. Using special analytical techniques (regression analy-sis), an analyst produced the following price–demand func-tion to model the data:

p(x) = 75 - 3x 1 … x … 20

118. Price–demand. A company manufactures “notebook” com-puters. Its marketing research department, using statisticaltechniques, collected the data shown in Table 9, where p is thewholesale price per computer at which x thousand computerscan be sold. Using special analytical techniques (regressionanalysis), an analyst produced the following price–demandfunction to model the data:

p(x) = 2,000 - 60x 1 … x … 25


x (Millions) p ($)

1 72

4 63

9 48

14 33

20 15

Plot the data points in Table 8, and sketch a graph of the price–demand function in the same coordinate system.What wouldbe the estimated price per chip for a demand of 7 millionchips? For a demand of 11 million chips?


x (Thousands) p ($)

1 1,940

8 1,520

16 1,040

21 740

25 500

Plot the data points in Table 9, and sketch a graph of theprice–demand function in the same coordinate system. Whatwould be the estimated price per computer for a demand of 11thousand computers? For a demand of 18 thousand computers?

Applications



119. Revenue.(A) Using the price–demand function

from Problem 117, write the company’s revenue functionand indicate its domain.

(B) Complete Table 10, computing revenues to the nearestmillion dollars.

p(x) = 75 - 3x 1 … x … 20

(C) Plot the points in part (B) and sketch a graph of theprofit function through these points.

122. Profit. The financial department for the company in Prob-lems 118 and 120 established the following cost function forproducing and selling x thousand “notebook” computers:

(A) Write a profit function for producing and selling xthousand “notebook” computers, and indicate thedomain of this function.

(B) Complete Table 13, computing profits to the nearestthousand dollars.

C(x) = 4,000 + 500x thousand dollars

TABLE 10 Revenue

x (Millions) R(x) (Million $)

1 72

4

8

12

16

20

(C) Plot the points from part (B) and sketch a graph of therevenue function through these points. Choose millionsfor the units on the horizontal and vertical axes.

120. Revenue.(A) Using the price–demand function

from Problem 118, write the company’s revenue functionand indicate its domain.

(B) Complete Table 11, computing revenues to the nearestthousand dollars.

p(x) = 2,000 - 60x 1 … x … 25

TABLE 11 Revenue

x (Thousands) R(x) (Thousand $)

1 1,940

5

10

15

20

25

(C) Plot the points from part (B) and sketch a graph of therevenue function through these points. Choose thou-sands for the units on the horizontal and vertical axes.

121. Profit. The financial department for the company inProblems 117 and 119 established the following cost func-tion for producing and selling x million memory chips:

(A) Write a profit function for producing and selling xmillion memory chips, and indicate its domain.

(B) Complete Table 12, computing profits to the nearestmillion dollars.

C(x) = 125 + 16x million dollars

TABLE 12 Profit


1 �69

4

8

12

16

20

TABLE 13 Profit

x (Thousands) P(x) (Thousand $)

1 �2,560

5

10

15

20

25

(C) Plot the points in part (B) and sketch a graph of theprofit function through these points.

123. Packaging. A candy box is to be made out of a piece of card-board that measures 8 by 12 inches. Equal-sized squares xinches on a side will be cut out of each corner, and then theends and sides will be folded up to form a rectangular box.

(A) Express the volume of the box V(x) in terms of x.

(B) What is the domain of the function V (determined bythe physical restrictions)?

(C) Complete Table 14.

TABLE 14 Volume

x V(x)

1

2

3

(D) Plot the points in part (C) and sketch a graph of thevolume function through these points.

124. Packaging. Refer to Problem 123.(A) Table 15 shows the volume of the box for some values

of x between 1 and 2. Use these values to estimate to one

TABLE 15 Volume

x V(x)

1.1 62.524

1.2 64.512

1.3 65.988

1.4 66.976

1.5 67.5

1.6 67.584

1.7 67.252


S e c t i o n 2 . 2 Elementary Functions: Graphs and Transformations 63

decimal place the value of x between 1 and 2 thatwould produce a box with a volume of 65 cubic inches.

(B) Describe how you could refine this table to estimate xto two decimal places.

(C) Carry out the refinement you described in part (B) andapproximate x to two decimal places.

125. Packaging. Refer to Problems 123 and 124.(A) Examine the graph of V(x) from Problem 123D and

discuss the possible locations of other values of x thatwould produce a box with a volume of 65 cubic inches.Construct a table like Table 15 to estimate any suchvalue to one decimal place.

(B) Refine the table you constructed in part (A) to providean approximation to two decimal places.

126. Packaging. A parcel delivery service will only deliver pack-ages with length plus girth (distance around) not exceeding108 inches. A rectangular shipping box with square ends xinches on a side is to be used.

(A) If the full 108 inches is to be used, express the volume ofthe box V(x) in terms of x.

(B) What is the domain of the function V (determined bythe physical restrictions)?

Length

x

x

Girth

(C) Complete Table 16.

TABLE 16 Volume

x V(x)

5

10

15

20

25

(D) Plot the points in part (C) and sketch a graph of thevolume function through these points.

127. Muscle contraction. In a study of the speed of muscle con-traction in frogs under various loads, noted British biophysi-cist and Nobel Prize winner A. W. Hill determined that theweight w (in grams) placed on the muscle and the speed ofcontraction v (in centimeters per second) are approximatelyrelated by an equation of the form

where a, b, and c are constants. Suppose that for a certain mus-cle, and Express v as a function of .Find the speed of contraction if a weight of 16 grams is placedon the muscle.

128. Politics. The percentage s of seats in the House of Represen-tatives won by Democrats and the percentage v of votes castfor Democrats (when expressed as decimal fractions) are re-lated by the equation

(A) Express v as a function of s, and find the percentage ofvotes required for the Democrats to win 51% of theseats.

(B) Express s as a function of v, and find the percentage ofseats won if Democrats receive 51% of the votes.

5v - 2s = 1.4 0 6 s 6 1, 0.28 6 v 6 0.68

wc = 90.a = 15, b = 1,

(w + a)(v + b) = c

ELEMENTARY FUNCTIONS: GRAPHS AND TRANSFORMATIONS� A Beginning Library of Elementary Functions� Vertical and Horizontal Shifts� Reflections, Stretches, and Shrinks� Piecewise-Defined Functions

The functions

all can be expressed in terms of the function as follows:

In this section we will see that the graphs of functions g, h, and k are closely relatedto the graph of function f. Insight gained by understanding these relationships will helpus analyze and interpret the graphs of many different functions.

g(x) = f(x) - 4 h(x) = f(x - 4) k(x) = -4f(x)

f(x) = x2

g(x) = x2- 4 h(x) = (x - 4)2 k(x) = -4x2

Section 2-2



� A Beginning Library of Elementary FunctionsAs you progress through this book, and most any other mathematics course beyondthis one, you will repeatedly encounter a relatively small list of elementary functions.We will identify these functions, study their basic properties, and include them in a li-brary of elementary functions (see the inside front cover). This library will becomean important addition to your mathematical toolbox and can be used in any courseor activity where mathematics is applied.

We begin by placing six basic functions in our library.

Basic Elementary Functions

Identity function

Square function

Cube function

Square root function

Cube root function

Absolute value function g(x) = ƒ x ƒ

p(x) =31x

n(x) = 1x

m(x) = x3

h(x) = x2

f(x) = x

DEFINITION

These elementary functions can be evaluated by hand for certain values of x andwith a calculator for all values of x for which they are defined.

Evaluating Basic Elementary Functions Evaluate each basic elementary func-tion at

(A)

(B)

Round any approximate values to four decimal places.

x = -12.75

x = 64

(A)

Use a calculator.Use a calculator.

(B)

Use a calculator.Use a calculator.Not a real number.Use a calculator.

g(-12.75) = ƒ -12.75 ƒ = 12.75

p(-12.75) =32-12.75 L -2.3362

n(-12.75) = 2-12.75

m(-12.75) = (-12.75)3 L -2,072.6719

h(-12.75) = (-12.75)2= 162.5625

f(-12.75) = -12.75

g(64) = ƒ 64 ƒ = 64

p(64) =3164 = 4

n(64) = 264 = 8

m(64) = 643= 262,144

h(64) = 642= 4,096

f(64) = 64SOLUTION

E X A M P L E 1

MATCHED PROBLEM 1 Evaluate each basic elementary function at

(A) (B)Round any approximate values to four decimal places.

x = -5.25x = 729



Figure 1 shows the graph, range, and domain of each of the basic elementaryfunctions.

Most computers and graphing calculators use ABS(x) to represent the absolute valuefunction. The following representation can also be useful:

ƒ x ƒ = 2x2

REMARK

�5

5�5

5

(A) Identity functionf (x) � x

Domain: RRange: R

�5

5�5

5

(F) Absolute value functiong (x) � �x �

Domain: RRange: [0, �)

f (x)

g (x)

�5

5�5

5

(B) Square functionh (x) � x2

Domain: RRange: [0, �)

h (x)

xx

x

�5

5�5

5

(D) Square root functionn (x) � �x

Domain: [0, �)Range: [0, �)

�5

5�5

5

(E) Cube root functionp (x) � �x

Domain: RRange: R

3

p(x)n (x)

xx

�5

5�5

5

(C) Cube functionm (x) � x3

Domain: RRange: R

m (x)

x

FIGURE 1 Some basic functions and their graphsNote: Letters used to designate these functions may vary from context to context; R is the set of allreal numbers.

Absolute Value In beginning algebra, absolute value is often interpreted as distance fromthe origin on a real number line (see Appendix A, Section A-1).

If then is the positive distance from the origin to x and if then x is thepositive distance from the origin to x. Thus,

ƒ x ƒ = e -x if x 6 0x if x Ú 0

x 7 0,-xx 6 0,

I N S I G H T

0 5�5�10 10

distance � 6 � �(�6) distance � 5

� Vertical and Horizontal ShiftsIf a new function is formed by performing an operation on a given function, then thegraph of the new function is called a transformation of the graph of the originalfunction. For example, graphs of both and are transfor-mations of the graph of y = f(x).

y = f(x + h)y = f(x) + k



Vertical and Horizontal Shifts

(A) How are the graphs of and related to the graph ofConfirm your answer by graphing all three functions simultaneously

in the same coordinate system.

(B) How are the graphs of and related to the graph ofConfirm your answer by graphing all three functions simultaneously

in the same coordinate system.y = ƒ x ƒ ?

y = ƒ x - 5 ƒy = ƒ x + 4 ƒ

y = ƒ x ƒ ?y = ƒ x ƒ - 5y = ƒ x ƒ + 4

(A) The graph of is the same as the graph of shifted upward4 units, and the graph of is the same as the graph of shifteddownward 5 units. Figure 2 confirms these conclusions. [It appears that the graphof is the graph of shifted up if k is positive and down ifk is negative.]

(B) The graph of is the same as the graph of shifted to the left4 units, and the graph of is the same as the graph of shiftedto the right 5 units. Figure 3 confirms these conclusions. [It appears that the graphof is the graph of shifted right if h is negative and left ifh is positive—the opposite of what you might expect.]

y = f(x)y = f(x + h)

y = ƒ x ƒy = ƒ x - 5 ƒ

y = ƒ x ƒy = ƒ x + 4 ƒ

y = f(x)y = f(x) + k

y = ƒ x ƒy = ƒ x ƒ - 5y = ƒ x ƒy = ƒ x ƒ + 4SOLUTION

E X A M P L E 2

y � �x �y � �x � � 4

y � �x � � 5

x

y

�5

�10

�10 5 10

10

FIGURE 2 Vertical shifts

y � �x �

x

y

�5

�5

�10

�10 5 10

5

10

y � �x � 4 �

y � �x � 5 �

FIGURE 3 Horizontal shifts

MATCHED PROBLEM 2 (A) How are the graphs of and related to the graph ofConfirm your answer by graphing all three functions simultaneously

in the same coordinate system.

(B) How are the graphs of and related to the graph ofConfirm your answer by graphing all three functions simultaneously in

the same coordinate system.

Comparing the graphs of with the graph of we see that thegraph of can be obtained from the graph of by verticallytranslating (shifting) the graph of the latter upward k units if k is positive and down-ward units if k is negative. Comparing the graphs of with the graphof we see that the graph of can be obtained from the graphy = f(x + h)y = f(x),

y = f(x + h)ƒ k ƒ

y = f(x)y = f(x) + ky = f(x),y = f(x) + k

y = 1x?y = 1x - 4y = 1x + 5

y = 1x ?y = 1x - 4y = 1x + 5

Let

(A) Graph for and 2 simultaneously in the same coor-dinate system. Describe the relationship between the graph of and the graph of for k any real number.

(B) Graph for and 2 simultaneously in the samecoordinate system. Describe the relationship between the graph of

and the graph of for h any real number.y = f(x + h)y = f(x)

h = -4, 0,y = f(x + h)

y = f(x) + ky = f(x)

k = -4, 0,y = f(x) + k

f(x) = x2.Explore & Discuss 1



� Reflections, Stretches, and ShrinksWe now investigate how the graph of is related to the graph of for different real numbers A.

y = f(x)y = Af(x)

Vertical and Horizontal Translations (Shifts) The graphs in Figure 4 are eitherhorizontal or vertical shifts of the graph of Write appropriate equationsfor functions H, G, M, and N in terms of f.

f(x) = x2.E X A M P L E 3

x

y

5

�5

�5 5x

y

5

�5

�5 5

f f

(A) (B)

H G M N

FIGURE 4 Vertical and horizontal shifts

of by horizontally translating (shifting) the graph of the latter h units to theleft if h is positive and units to the right if h is negative.ƒ h ƒ

y = f(x)

Functions H and G are vertical shifts given by

Functions M and N are horizontal shifts given by

M(x) = (x + 2)2 N(x) = (x - 3)2

H(x) = x2+ 2 G(x) = x2

- 4

SOLUTION

MATCHED PROBLEM 3 The graphs in Figure 5 are either horizontal or vertical shifts of the graph ofWrite appropriate equations for functions H, G, M, and N in terms of f.f(x) =

31x.

5

�5

�5 5

5

�5

�5 5

y

x

y

f

f

x

(A) (B)

H

G

M

N

FIGURE 5 Vertical and horizontal shifts



Comparing to we see that the graph of can be ob-tained from the graph of by multiplying each ordinate value of the latter byA. The result is a vertical stretch of the graph of if a vertical shrinkof the graph of if and a reflection in the x axis if If Ais a negative number other than then the result is a combination of a reflectionin the x axis and either a vertical stretch or a vertical shrink.

-1,A = -1.0 6 A 6 1,y = f(x)

A 7 1,y = f(x)y = f(x)

y = Af(x)y = f(x),y = Af(x)

(A) Graph for and simultaneously in the same coordinatesystem.

(B) Graph for and simultaneously in the same coor-dinate system.

(C) Describe the relationship between the graph of and the graphof for A any real number.G(x) = Ax2

h(x) = x2

-14A = -1, -4,y = Ax2

14A = 1, 4,y = Ax2Explore & Discuss 2

Reflections, Stretches, and Shrinks

(A) How are the graphs of and related to the graph of Confirm your answer by graphing all three functions simultaneously in the samecoordinate system.

(B) How is the graph of related to the graph of Confirm youranswer by graphing both functions simultaneously in the same coordinatesystem.

y = ƒ x ƒ ?y = -2 ƒ x ƒ

y = ƒ x ƒ ?y = 0.5 ƒ x ƒy = 2 ƒ x ƒ

(A) The graph of is a vertical stretch of the graph of by a factor of2, and the graph of is a vertical shrink of the graph of by afactor of 0.5. Figure 6 confirms this conclusion.

(B) The graph of is a reflection in the x axis and a vertical stretch of thegraph of . Figure 7 confirms this conclusion.y = ƒ x ƒ

y = -2 ƒ x ƒ

y = ƒ x ƒy = 0.5 ƒ x ƒ

y = ƒ x ƒy = 2 ƒ x ƒSOLUTION

E X A M P L E 4

x

y

�5

�5

�10

�10 5 10

5

10

y � �x �

y � 0.5�x �

y � 2�x �

FIGURE 6 Vertical stretch andshrink

�5

�5

�10

�10 5 10

5

10

y � �x �

x

y

y � �2�x �

FIGURE 7 Reflection and vertical stretch

MATCHED PROBLEM 4 (A) How are the graphs of and related to the graph of Con-firm your answer by graphing all three functions simultaneously in the samecoordinate system.

(B) How is the graph of related to the graph of Confirm youranswer by graphing both functions in the same coordinate system.

The various transformations considered above are summarized in the following boxfor easy reference:

y = x?y = -0.5x

y = x?y = 0.5xy = 2x



GRAPH TRANSFORMATIONS

Vertical Translation:

Horizontal Translation:

Reflection:in the x axis.

Vertical Stretch and Shrink:

y = Af(x) µA 7 1 Stretch graph of y = f(x) vertically

by multiplying each ordinate value by A.0 6 A 6 1 Shrink graph of y = f(x) vertically

by multiplying each ordinate value by A.

y = -f(x) Reflect the graph of y = f(x)

y = f(x + h) eh 7 0 Shift graph of y = f(x) left h units.h 6 0 Shift graph of y = f(x) right ƒ h ƒ units.

y = f(x) + k ek 7 0 Shift graph of y = f(x) up k units.k 6 0 Shift graph of y = f(x) down ƒ k ƒ units.

SUMMARY

Use a graphing calculator to explore the graph of for vari-ous values of the constants A, h, and k. Discuss how the graph of

is related to the graph of y = x2.y = A(x + h)2+ k

y = A(x + h)2+ kExplore & Discuss 3

Combining Graph Transformations Discuss the relationship between thegraphs of and Confirm your answer by graphing bothfunctions simultaneously in the same coordinate system.

y = ƒ x ƒ .y = - ƒ x - 3 ƒ + 1

The graph of is a reflection in the x axis, a horizontal translationof 3 units to the right, and a vertical translation of 1 unit upward of the graph of

Figure 8 confirms this description.y = ƒ x ƒ .

y = - ƒ x - 3 ƒ + 1SOLUTION

E X A M P L E 5

y � �x �

x

y

�5

5�5

5

y � ��x � 3� � 1

FIGURE 8 Combined transformations

MATCHED PROBLEM 5 The graph of in Figure 9 on the next page involves a reflection and a trans-lation of the graph of Describe how the graph of function G is related to thegraph of and find an equation of the function G.y = x3

y = x3.y = G(x)



� Piecewise-Defined FunctionsEarlier we noted that the absolute value of a real number x can be defined as

Notice that this function is defined by different rules for different parts of its domain.Functions whose definitions involve more than one rule are called piecewise-definedfunctions. Graphing one of these functions involves graphing each rule over the ap-propriate portion of the domain (Fig. 10). In Figure 10C, notice that an open dot isused to show that the point is not part of the graph and a solid dot is used toshow that (0, 2) is part of the graph.

As the next example illustrates, piecewise-defined functions occur naturally inmany applications.

(0, -2)

ƒ x ƒ = e -x if x 6 0x if x Ú 0

5�5

5

x

yG

FIGURE 9 Combined transformations

x

y

�5

5�5

5

x

y

�5

5�5

5

x

y

�5

5�5

5

(A) y � x2 � 2 (B) y � 2 � x2 (C) y � x2 � 2 if x � 0

2 � x2 if x � 0

FIGURE 10 Graphing a piecewise-defined function

Natural Gas Rates Easton Utilities uses the rates shown in Table 1 to computethe monthly cost of natural gas for each customer.Write a piecewise definition for thecost of consuming x CCF (cubic hundred feet) of natural gas and graph the function.

E X A M P L E 6

TABLE 1 Charges per Month

$0.7866 per CCF for the first 5 CCF

$0.4601 per CCF for the next 35 CCF

$0.2508 per CCF for all over 40 CCF

If C(x) is the cost, in dollars, of using x CCF of natural gas in one month, then the firstline of Table 1 implies that

C(x) = 0.7866x if 0 … x … 5

SOLUTION



Note that is the cost of 5 CCF. If then represents theamount of gas that cost $0.4601 per CCF, represents the cost of this gas,and the total cost is

If then

where the cost of the first 40 CCF. Combining all these equations,we have the following piecewise definition for C(x):

To graph C, first note that each rule in the definition of C represents a transforma-tion of the identity function Graphing each transformation over the indi-cated interval produces the graph of C shown in Figure 11.

f(x) = x.

C(x) = µ0.7866x if 0 … x … 53.933 + 0.46501 (x - 5) if 5 6 x … 4020.0365 + 0.2508 (x - 40) if 40 6 x

20.0365 = C(40),

C(x) = 20.0365 + 0.2508(x - 40)

x 7 40,

C(x) = 3.933 + 0.4601(x - 5)

0.4601(x - 5)x - 55 6 x … 40,C(5) = 3.933

2010 4030 50 60x

C(x)

$10

$20

$30

FIGURE 11 Cost of purchasing x CCF of natural gas

MATCHED PROBLEM 6 Natural Gas Rates Trussville Utilities uses the rates shown in Table 2 to computethe monthly cost of natural gas for residential customers.Write a piecewise definitionfor the cost of consuming x CCF of natural gas and graph the function.

TABLE 2 Charges per Month

$0.7675 per CCF for the first 50 CCF

$0.6400 per CCF for the next 150 CCF

$0.6130 per CCF for all over 200 CCF

1. (A)

(B)

2. (A) The graph of is the same as the graph of shifted upward5 units, and the graph of is the same as the graph of shifteddownward 4 units. The figure confirms these conclusions.

y = 1xy = 1x - 4y = 1xy = 1x + 5

p(-5.25) = -1.7380, g(-5.25) = 5.25

m(-5.25) = -144.7031, n(-5.25) is not a real number,

f(-5.25) = -5.25, h(-5.25) = 27.5625,

n(729) = 27, p(729) = 9, g(729) = 729

f(729) = 729, h(729) = 531,441, m(729) = 387,420,489,Answers to Matched Problems

�5

�5

�10

�10 5

5

10 y � �x � 5

y � �x � 4

y � �x

x

y



(B) The graph of is the same as the graph of shifted to the left 5units, and the graph of is the same as the graph of shifted tothe right 4 units. The figure confirms these conclusions.

y = 1xy = 1x - 4y = 1xy = 1x + 5

3.

4. (A) The graph of is a vertical stretch of the graph of and the graph ofis a vertical shrink of the graph of The figure confirms these

conclusions.y = x.y = 0.5x

y = x,y = 2x

N(x) = 13 x - 3M(x) = 13 x + 2,G(x) = 13 x - 2,H(x) = 13 x + 3,

(B) The graph of is a vertical shrink and a reflection in the x axis of the graphof The figure confirms this conclusion.y = x.

y = -0.5x

�5

�10

�10 5

5

10

�5

y � �x � 5

y � �x � 4

y � �x

x

y

x

C(x)

200

100

0300100 200

�5

�5

�10

�10 5 10

5

10

y � xy � 2x

y � 0.5x

x

y

x

y

�5

�5

�10

�10 5 10

5

10y � x

y � �0.5x

5. The graph of function G is a reflection in the x axis and a horizontal translation of 2 unitsto the left of the graph of An equation for G is

6. C(x) = µ0.7675x if 0 … x … 5038.375 + 0.64 (x - 50) if 50 6 x … 200134.375 + 0.613 (x - 200) if 200 6 x

G(x) = -(x + 2)3.y = x3.



A Without looking back in the text, indicate the domain and rangeof each of the functions in Problems 1–8.

1. 2.

3. 4.

5. 6.

7. 8.

Graph each of the functions in Problems 9–20 using the graphsof functions f and g below.

s(x) = 5 31xr(x) = -x3

n(x) = -0.1x2m(x) = 3 ƒ x ƒ

k(x) = 41xh(x) = -0.61x

g(x) = -0.3xf(x) = 2x

Exercise 2-2

31. 32.

�5

5�5

5

�5

5�5

5

xx

g(x)

xx

f (x)

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

B In Problems 21–28, indicate verbally how the graph of eachfunction is related to the graph of one of the six basic functionsin Figure 1 on page 65. Sketch a graph of each function.

21.

22.

23.

24.

25.

26.

27.

28.

Each graph in Problems 29–36 is the result of applying a sequenceof transformations to the graph of one of the six basic functions inFigure 1 on page 65. Identify the basic function and describe thetransformation verbally.Write an equation for the given graph.

29. 30.

m(x) = -0.4x2

h(x) = -3 ƒ x ƒ

g(x) = -6 +31x

f(x) = 7 - 1x

m(x) = (x + 3)2+ 4

f(x) = (x - 4)2- 3

h(x) = - ƒ x - 5 ƒ

g(x) = - ƒ x + 3 ƒ

y = 2f(x)y = 0.5g(x)

y = -g(x)y = -f(x)

y = f(x) + 3y = g(x) - 3

y = f(x + 3)y = g(x - 3)

y = g(x - 1)y = f(x + 2)

y = g(x) - 1y = f(x) + 2

x

y

�5

5�5

5

x

y

�5

5�5

5

y

�5

5�5

5

x

y

�5

5�5

5

x

x

y

�5

5�5

5

x

y

�5

5�5

5

x

y

�5

5�5

5

33. 34.

35. 36.

x

y

�5

5�5

5

In Problems 37–42, the graph of the function g is formed by applying the indicated sequence of transformations to the givenfunction f. Find an equation for the function g and graph gusing and .

37. The graph of is shifted 2 units to the rightand 3 units down.

38. The graph of is shifted 3 units to the leftand 2 units up.

39. The graph of is reflected in the x axisand shifted to the left 3 units.

40. The graph of is reflected in the x axisand shifted to the right 1 unit.

41. The graph of is reflected in the x axisand shifted 2 units to the right and down 1 unit.

42. The graph of is reflected in the x axis and shifted to the left 2 units and up 4 units.

f(x) = x2

f(x) = x3

f(x) = ƒ x ƒ

f(x) = ƒ x ƒ

f(x) =31x

f(x) = 1x

-5 … y … 5-5 … x … 5



Graph each function in Problems 43–48.

43.

44.

45.

46.

47.

48.

C Each of the graphs in Problems 49–54 involves a reflection in thex axis and/or a vertical stretch or shrink of one of the basic func-tions in Figure 1 on page 65. Identify the basic function, and de-scribe the transformation verbally.Write an equation for the givengraph.

49. 50.

h(x) = c 4x + 20 if 0 … x … 202x + 60 if 20 6 x … 100-x + 360 if x 7 100

h(x) = c 2x if 0 … x … 20x + 20 if 20 6 x … 400.5x + 45 if x 7 40

h(x) = b10 + 2x if 0 … x … 2040 + 0.5x if x 7 20

h(x) = b5 + 0.5x if 0 … x … 10-10 + 2x if x 7 10

g(x) = b x + 1 if x 6 -12 + 2x if x Ú -1

f(x) = b2 - 2x if x 6 2x - 2 if x Ú 2

x

y

�5

5�5

5

x

y

�5

5�5

5

51. 52.

x

y

�5

5�5

5

x

y

�5

5�5

5

x

y

�5

5�5

5

x

y

�5

5�5

5

53. 54.

Changing the order in a sequence of transformations maychange the final result. Investigate each pair of transformationsin Problems 55–60 to determine if reversing their order canproduce a different result. Support your conclusions with spe-cific examples and/or mathematical arguments.

55. Vertical shift; horizontal shift

56. Vertical shift; reflection in y axis

57. Vertical shift; reflection in x axis

58. Vertical shift; vertical stretch

59. Horizontal shift; reflection in y axis

60. Horizontal shift; vertical shrink

Applications

61. Price–demand. A retail chain sells CD players. The retailprice p(x) (in dollars) and the weekly demand x for a par-ticular model are related by

(A) Describe how the graph of function p can be ob-tained from the graph of one of the basic functionsin Figure 1 on page 65.

(B) Sketch a graph of function p using part (A) as an aid.

62. Price–supply. The manufacturers of the CD players inProblem 61 are willing to supply x players at a price of p(x)as given by the equation


(B) Sketch a graph of function p using part (A) as an aid.

p(x) = 41x 9 … x … 289

p(x) = 115 - 41x 9 … x … 289

63. Hospital costs. Using statistical methods, the financialdepartment of a hospital arrived at the cost equation

where C(x) is the cost in dollars for handling x cases permonth.

(A) Describe how the graph of function C can beobtained from the graph of one of the basicfunctions in Figure 1 on page 65.

(B) Sketch a graph of function C using part (A) and agraphing calculator as aids.

64. Price–demand. A company manufactures and sells in-lineskates. Its financial department has established theprice–demand function

p(x) = 190 - 0.013(x - 10)2 10 … x … 100

C(x) = 0.00048(x - 500)3+ 60,000 100 … x … 1,000



where p(x) is the price at which x thousand pairs of skatescan be sold.


(B) Sketch a graph of function p using part (A) and agraphing calculator as aids.

65. Electricity rates. Table 3 shows the electricity rates chargedby Monroe Utilities in the summer months. The base is afixed monthly charge, independent of the kWh (kilowatt-hours) used during the month.

(A) Write a piecewise definition of the monthly chargeS(x) for a customer who uses x kWh in a summermonth.

(B) Graph S(x).

TABLE 3 Summer (July–October)

TABLE 5 Kansas State Income Tax

TABLE 4 Winter (November–June)

66. Electricity rates. Table 4 shows the electricity rates chargedby Monroe Utilities in the winter months.

(A) Write a piecewise definition of the monthly chargeW(x) for a customer who uses x kWh in a wintermonth.

TABLE 6 Kansas State Income Tax

(A) Write a piecewise definition for the tax due T(x) onan income of x dollars.

(B) Graph T(x).

(C) Find the tax due on a taxable income of $20,000. Of$35,000.

(D) Would it be better for a married couple in Kansaswith two equal incomes to file jointly or sepa-rately? Discuss.

69. Physiology. A good approximation of the normal weight ofa person 60 inches or taller but not taller than 80 inches isgiven by where x is height in inchesand is weight in pounds.w(x)

w(x) = 5.5x - 220,

(B) Graph W(x).

67. State income tax. Table 5 shows a recent state income taxschedule for married couples filing a joint return in thestate of Kansas.

(A) Write a piecewise definition for the tax due T(x) onan income of x dollars.

(B) Graph T(x).

(C) Find the tax due on a taxable income of $40,000. Of$70,000.

68. State income tax. Table 6 shows a recent state income taxschedule for individuals filing a return in the state ofKansas.

(A) Describe how the graph of function can be ob-tained from the graph of one of the basic functionsin Figure 1, page 65.

(B) Sketch a graph of function using part (A) as an aid.

70. Physiology. The average weight of a particular species ofsnake is given by , , where xis length in meters and w(x) is weight in grams.

(A) Describe how the graph of function w can be ob-tained from the graph of one of the basic functionsin Figure 1, page 65.

(B) Sketch a graph of function w using part (A) as an aid.

71. Safety research. Under ideal conditions, if a person drivinga vehicle slams on the brakes and skids to a stop, the speedof the vehicle v(x) (in miles per hour) is given approxi-mately by , where x is the length of skidmarks (in feet) and C is a constant that depends on theroad conditions and the weight of the vehicle. For a par-ticular vehicle, and .

(A) Describe how the graph of function v can be ob-tained from the graph of one of the basic functionsin Figure 1, page 65.

(B) Sketch a graph of function v using part (A) as an aid.

4 … x … 144v(x) = 7.081x

v(x) = C1x

0.2 … x … 0.8w(x) = 463x3

w

w

SCHEDULE II—SINGLE, HEAD OF HOUSEHOLD, OR MARRIED FILING SEPARATE

If taxable income is

Over But Not Over Tax Due Is

$0 $15,000 3.50% of taxable income

$15,000 $30,000 $525 plus 6.25% ofexcess over $15,000

$30,000 $1,462.50 plus 6.45% ofexcess over $30,000

Base charge, $8.50

First 700 kWh or less at 0.0650/kWh

Over 700 kWh at 0.0530/kWh

Base charge, $8.50

First 700 kWh or less at 0.0650/kWh

Over 700 kWh at 0.0900/kWh

SCHEDULE I—MARRIED FILING JOINT

If taxable income is

Over But Not Over Tax Due Is

$0 $30,000 3.50% of taxable income

$30,000 $60,000 $1,050 plus 6.25% of excess over $30,000

$60,000 $2,925 plus 6.45% ofexcess over $60,000



QUADRATIC FUNCTIONS� Quadratic Functions, Equations, and Inequalities� Properties of Quadratic Functions and Their Graphs� Applications� More General Functions: Polynomial and Rational Functions

If the degree of a linear function is increased by one, we obtain a second-degree func-tion, usually called a quadratic function, another basic function that we will need inour library of elementary functions. We will investigate relationships betweenquadratic functions and the solutions to quadratic equations and inequalities. Otherimportant properties of quadratic functions will also be investigated, includingmaximum and minimum properties. We will then be in a position to solve importantpractical problems such as finding production levels that will produce maximumrevenue or maximum profit.

� Quadratic Functions, Equations, and InequalitiesThe graph of the square function is shown in Figure 1. Notice that thegraph is symmetric with respect to the y axis and that (0, 0) is the lowest point onthe graph. Let’s explore the effect of applying a sequence of basic transformations tothe graph of h.

Indicate how the graph of each function is related to the graph of the function. Find the highest or lowest point, whichever exists, on each graph.

(A)

(B)

(C)

(D)

Graphing the functions in Explore–Discuss 1 produces figures similar in shape tothe graph of the square function in Figure 1. These figures are called parabolas. Thefunctions that produced these parabolas are examples of the important class ofquadratic functions, which we now define.

n(x) = -3(x + 1)2- 1 = -3x2

- 6x - 4

m(x) = -(x - 4)2+ 8 = -x2

+ 8x - 8

g(x) = 0.5(x + 2)2+ 3 = 0.5x2

+ 2x + 5

f(x) = (x - 3)2- 7 = x2

- 6x + 2

h(x) = x2

h(x) = x2

Section 2-3

Quadratic Functions If a, b, and c are real numbers with , then the function

Standard form

is a quadratic function and its graph is a parabola.

f(x) = ax2+ bx + c

a Z 0DEFINITION

If x is any real number, then is also a real number. According to the agree-ment on domain and range in Section 2-1, the domain of a quadratic function is R, the setof real numbers.

ax2+ bx + c

I N S I G H T

x

h(x)

�5

5�5

5

FIGURE 1 Square function h(x) = x2

72. Learning. A production analyst has found that on the av-erage it takes a new person T(x) minutes to perform a par-ticular assembly operation after x performances of theoperation, where .T(x) = 10 -

31x, 0 … x … 125

(A) Describe how the graph of function T can be ob-tained from the graph of one of the basic functionsin Figure 1, page 65.

(B) Sketch a graph of function T using part (A) as an aid.

We will discuss methods for determining the range of a quadratic function later inthis section. Typical graphs of quadratic functions are illustrated in Figure 2.

Explore & Discuss 1


S e c t i o n 2 . 3 Quadratic Functions 77

Intercepts, Equations, and Inequalities

(A) Sketch a graph of in a rectangular coordinate system.

(B) Find x and y intercepts algebraically to four decimal places.

(C) Graph in a standard viewing window.

(D) Find the x and y intercepts to four decimal places using TRACE and zero on yourgraphing calculator.

(E) Solve the quadratic inequality graphically to four decimalplaces using the results of parts (A) and (B) or (C) and (D).

(F) Solve the equation graphically to four decimal places usingINTERSECT on your graphing calculator.

-x2+ 5x + 3 = 4

-x2+ 5x + 3 Ú 0

f(x) = -x2+ 5x + 3

f(x) = -x2+ 5x + 3

x x

h (x)

�10

5�5

10

(A) f (x) � x2 � 4 (B) g (x) � 3x2 � 12x � 14 (C) h(x) � 3 � 2x � x2

g (x)

�10

5�5

10

x

�10

5�5

10

f(x)

FIGURE 2 Graphs of quadratic functions

E X A M P L E 1

SOLUTION (A) Hand-sketching a graph of f :

The x intercepts of a linear function can be found by solving the linear equationfor x, (see Section 1-2). Similarly, the x intercepts of a quadratic

function can be found by solving the quadratic equation for x,Several methods for solving quadratic equations are discussed in Appendix A, Section A-7.The most popular of these is the quadratic formula.

If , then

, provided b2- 4ac Ú 0x =

-b ; 2b2- 4ac

2a

ax2+ bx + c = 0, a Z 0

a Z 0.y = ax2+ bx + c = 0

m Z 0y = mx + b = 0

I N S I G H T

5

x

f (x)

5�5

10

�10 10

(B) Finding intercepts algebraically:

y intercept:x intercepts:

Quadratic equation-x2+ 5x + 3 = 0

f(x) = 0 f(0) = -(0)2

+ 5(0) + 3 = 3

x y

�1 �3

0 3

1 7

2 9

3 9

4 7

5 3

6 �3



(D) Finding intercepts graphically using a graphing calculator:

�10

�10

10

10

�10

�10

10

10

y intercept: 3x intercept: -0.5414x intercept: -0.5414

-x2+ 5x + 3 = 4 at x = 4.7913-x2

+ 5x + 3 = 4 at x = 0.2087

(E) Solving graphically: The quadratic inequality

holds for those values of x for which the graph of in the figures in parts (A) and (C) is at or above the x axis. This happens for xbetween the two x intercepts [found in part (B) or (D)], including the two x intercepts. Thus, the solution set for the quadratic inequality is

or .

(F) Solving the equation using a graphing calculator:-x2+ 5x + 3 = 4

[-0.5414, 5 .5414]-0.5414 … x … 5.5414

f(x) = -x2+ 5x + 3

-x2+ 5x + 3 Ú 0

-x2+ 5x + 3 Ú 0

�10

�10

10

10

Quadratic formula (see Appendix A-7)

(C) Graphing in a graphing calculator:

=

-5 ; 237-2

= -0.5414 or 5.5414

x =

-(5) ; 252- 4(-1)(3)

2(-1)

x =

-b ; 2b2- 4ac

2a

MATCHED PROBLEM 1 (A) Sketch a graph of in a rectangular coordinate system.

(B) Find x and y intercepts algebraically to four decimal places.

(C) Graph in a standard viewing window.

(D) Find the x and y intercepts to four decimal places using TRACE and the zero com-mand on your graphing calculator.

(E) Solve graphically to four decimal places using the results ofparts (A) and (B) or (C) and (D).

(F) Solve the equation graphically to four decimal placesusing an appropriate built-in routine in your graphing calculator.

2x2- 5x - 5 = -3

2x2- 5x - 5 Ú 0

g(x) = 2x2- 5x - 5

g(x) = 2x2- 5x - 5



� Properties of Quadratic Functions and Their GraphsMany useful properties of the quadratic function can be uncovered by transforming

into the vertex form*

The process of completing the square (see Appendix A-7) is central to the transfor-mation.We illustrate the process through a specific example and then generalize theresults.

Consider the quadratic function given by

(1)

We use completing the square to transform this function into vertex form:

Thus,

(2)

If then and For any other value of x, the nega-tive number is added to 8, making it smaller. (Think about this.) Therefore,

is the maximum value of f(x) for all x—a very important result! Furthermore, if wechoose any two x values that are the same distance from 4, we will obtain the samefunction value. For example, and are each one unit from and theirfunction values are

Thus, the vertical line is a line of symmetry.That is, if the graph of equation (1)is drawn on a piece of paper and the paper is folded along the line then thetwo sides of the parabola will match exactly.All these results are illustrated by graph-ing equations (1) and (2) and the line simultaneously in the same coordinatesystem (Fig. 3).

From the preceding discussion, we see that as x moves from left to right, f(x) isincreasing on and decreasing on and that f(x) can assume no valuegreater than 8. Thus,

Range of f : y … 8 or (- q , 8]

[4, q),(-q, 4],

x = 4

x = 4,x = 4

f(5) = -2(5 - 4)2+ 8 = 6

f(3) = -2(3 - 4)2+ 8 = 6

x = 4x = 5x = 3

f(4) = 8

-2(x - 4)2f(4) = 8.-2(x - 4)2

= 0x = 4,

f(x) = -2(x - 4)2+ 8

= -2(x - 4)2+ 8

= -2(x2- 8x + 16) - 24 + 32

= -2(x2- 8x + ?) - 24

= -2(x2- 8x) - 24

f(x) = -2x2+ 16x - 24

f(x) = -2x2+ 16x - 24

f(x) = a(x - h)2+ k

f(x) = ax2+ bx + c a Z 0

Add 16 to complete the square in-side the parentheses. Because ofthe outside the parentheses, wehave actually added , so wemust add 32 to the outside.The transformation is complete andcan be checked by multiplying out.

-32-2

x

f (x)

5

5

10

�10

10

�5

Maximum: f (4) � 8

Line of symmetry: x � 4

f (x) � �2x2 � 16x � 24� �2(x � 4)2 � 8

FIGURE 3 Graph of a quadraticfunction

Factor the coefficient of out ofthe first two terms.

x2

How many x intercepts can the graph of a quadratic function have? How manyy intercepts? Explain your reasoning.

Explore & Discuss 2

* This terminology is not universally agreed upon. Some call this the standard form.



Note the important results we have obtained from the vertex form of the qua-dratic function f :

• The vertex of the parabola

• The axis of the parabola

• The maximum value of f(x)

• The range of the function f

• The relationship between the graph of and the graph of

Now, let us explore the effects of changing the constants a, h, and k on the graphof f(x) = a(x - h)2

+ k.

f(x) = -2x2+ 16x - 24

g(x) = x2

In general, the graph of a quadratic function is a parabola with line of symmetry par-allel to the vertical axis. The lowest or highest point on the parabola, whichever ex-ists, is called the vertex. The maximum or minimum value of a quadratic functionalways occurs at the vertex of the parabola.The line of symmetry through the vertexis called the axis of the parabola. In the example above, is the axis of theparabola and (4, 8) is its vertex.

x = 4

Applying the graph transformation properties discussed in Section 2-2 to the transformedequation,

we see that the graph of is the graph of verticallystretched by a factor of 2, reflected in the x axis, and shifted to the right 4 units and up 8 units, as shown in Figure 4.

g(x) = x2f(x) = -2x2+ 16x - 24

= -2(x - 4)2+ 8

f(x) = -2x2+ 16x - 24

I N S I G H T

� �2x2 � 16x � 24� �2(x � 4)2 � 8

f (x)g (x) � x2

x

y

5�5

5

10

�10

�10 10

�5

FIGURE 4 Graph of f is the graph of g transformed

(A) Let and Graph for and 3simultaneously in the same coordinate system. Explain the effect of chang-ing k on the graph of f.

(B) Let and Graph for and 5simultaneously in the same coordinate system. Explain the effect of chang-ing h on the graph of f.

(C) Let and Graph for and 3simultaneously in the same coordinate system. Graph function f for

a = 0.25, 1,f(x) = a(x - h)2+ kk = -2.h = 5

h = -4, 0,f(x) = a(x - h)2+ kk = 2.a = 1

k = -4, 0,f(x) = a(x - h)2+ kh = 5.a = 1Explore & Discuss 3



and simultaneously in the same coordinate system. Explainthe effect of changing a on the graph of f.

(D) Discuss parts (A)–(C) using a graphing calculator and a standard viewingwindow.

The preceding discussion is generalized for all quadratic functions in the follow-ing summary:

-0.25a = 1, -1,

PROPERTIES OF A QUADRATIC FUNCTION AND ITS GRAPH

Given a quadratic function and the vertex form obtained by completing thesquare

Standard form

Vertex form

we summarize general properties as follows:

1. The graph of f is a parabola:

2. Vertex: (h, k) (parabola increases on one side of the vertex and decreaseson the other)

3. Axis (of symmetry): (parallel to y axis)

4. is the minimum if and the maximum if

5. Domain: All real numbersRange: if or if

6. The graph of f is the graph of translated horizontally h units andvertically k units.

g(x) = ax2

a 7 0[k, q)a 6 0(- q , k]

a 6 0a 7 0f(h) = k

x = h

= a(x - h)2+ k

f(x) = ax2+ bx + c a Z 0

SUMMARY

x x

f (x) f (x)

Vertex (h, k)

Vertex (h, k)

Min f (x)

Max f (x)

Axisx � h

Axisx � h

k

k

h h

a � 0Opens upward

a � 0Opens downward

Analyzing a Quadratic Function Given the quadratic function

(A) Find the vertex form for f.

(B) Find the vertex and the maximum or minimum. State the range of f.

(C) Describe how the graph of function f can be obtained from the graphof using transformations discussed in Section 2-2.

(D) Sketch a graph of function f in a rectangular coordinate system.

(E) Graph function f using a suitable viewing window.

(F) Find the vertex and the maximum or minimum graphically using the minimumcommand. State the range of f.

g(x) = x2

f(x) = 0.5x2- 6x + 21

E X A M P L E 2



(E) Graph in a graphing calculator:

(A) Complete the square to find the vertex form:

(B) From the vertex form, we see that and Thus, vertex: (6, 3); mini-mum: range: or

(C) The graph of is the same as the graph of vertically shrunk by a factor of 0.5, and shifted to the right 6 units and up3 units.

(D) Graph in a rectangular coordinate system:

g(x) = x2f(x) = 0.5(x - 6)2+ 3

[3, q).y Ú 3f(6) = 3;k = 3.h = 6

= 0.5(x - 6)2+ 3

= 0.5(x2- 12x + 36) + 21 - 18

= 0.5(x2- 12x + ?) + 21

f(x) = 0.5x2- 6x + 21

SOLUTION

x

f (x)

5 10

5

10

�10

�10

10

10

(F) Finding the vertex, minimum, and range graphically using a graphing calculator:

�10

�10

10

10

Vertex: (6, 3); minimum: range: or [3, q).y Ú 3f(6) = 3;

MATCHED PROBLEM 2 Given the quadratic function

(A) Find the vertex form for f.

(B) Find the vertex and the maximum or minimum. State the range of f.

(C) Describe how the graph of function f can be obtained from the graph ofusing transformations discussed in Section 2-2.

(D) Sketch a graph of function f in a rectangular coordinate system.

(E) Graph function f using a suitable viewing window.

(F) Find the vertex and the maximum or minimum graphically using the minimumcommand. State the range of f.

g(x) = x2

f(x) = -0.25x2- 2x + 2



APPLICATIONS

Maximum Revenue This is a continuation of Example 6 in Section 2-1. Recall thatthe financial department in the company that produces a digital camera arrived at thefollowing price–demand function and the corresponding revenue function:

Price–demand functionRevenue function

where p(x) is the wholesale price per camera at which x million cameras can be soldand R(x) is the corresponding revenue (in million dollars). Both functions havedomain

(A) Find the output to the nearest thousand cameras that will produce the maxi-mum revenue. What is the maximum revenue to the nearest thousand dollars?Solve the problem algebraically by completing the square.

(B) What is the wholesale price per camera (to the nearest dollar) that produces themaximum revenue?

(C) Graph the revenue function using an appropriate viewing window.

(D) Find the output to the nearest thousand cameras that will produce the maxi-mum revenue. What is the maximum revenue to the nearest thousand dollars?Solve the problem graphically using the maximum command.

1 … x … 15.

R(x) = xp(x) = x(94.8 - 5x)

p(x) = 94.8 - 5x

(A) Algebraic solution:

The maximum revenue of 449.352 million dollars ($449,352,000) occurs whenmillion cameras (9,480,000 cameras).

(B) Finding the wholesale price per camera: Use the price–demand function for anoutput of 9.480 million cameras:

(C) Graph in a graphing calculator: = $47

p(9.480) = 94.8 - 5(9.480)

p(x) = 94.8 - 5x

x = 9.480

= -5(x - 9.48)2+ 449.352

= -5(x2- 18.96x + 89.8704) + 449.352

= -5(x2 - 18.96x + ?)

= -5x2+ 94.8x

R(x) = x(94.8 - 5x)SOLUTION

E X A M P L E 3

0

1

500

15

0

1

500

15

An output of 9.480 million cameras (9,480,000 cameras) will produce a maxi-mum revenue of 449.352 million dollars ($449,352,000).

(D) Graphical solution using a graphing calculator:



MATCHED PROBLEM 3 The financial department in Example 3, using statistical and analytical techniques(see Matched Problem 7 in Section 2-1), arrived at the cost function

Cost function

where C(x) is the cost (in million dollars) for manufacturing and selling x millioncameras.

(A) Using the revenue function from Example 3 and the preceding cost function,write an equation for the profit function.

(B) Find the output to the nearest thousand cameras that will produce the maximumprofit. What is the maximum profit to the nearest thousand dollars? Solve theproblem algebraically by completing the square.

(C) What is the wholesale price per camera (to the nearest dollar) that produces themaximum profit?

(D) Graph the profit function using an appropriate viewing window.

(E) Find the output to the nearest thousand cameras that will produce the maximumprofit. What is the maximum profit to the nearest thousand dollars? Solve theproblem graphically using the maximum routine.

C(x) = 156 + 19.7x

Break-Even Analysis Use the revenue function from Example 3 and the costfunction from Matched Problem 3:

Revenue functionCost function

Both have domain

(A) Sketch the graphs of both functions in the same coordinate system.

(B) Break-even points are the production levels at which Find thebreak-even points algebraically to the nearest thousand cameras.

(C) Plot both functions simultaneously in the same viewing window.

(D) Use INTERSECT to find the break-even points graphically to the nearest thousandcameras.

(E) Recall that a loss occurs if and a profit occurs if For what outputs (to the nearest thousand cameras) will a loss occur? A profit?

R(x) 7 C(x).R(x) 6 C(x)

R(x) = C(x).

1 … x … 15.

C(x) = 156 + 19.7x

R(x) = x(94.8 - 5x)

E X A M P L E 4

(A) Sketch of functions:SOLUTION

x

C(x)R(x)

5 10 15

250

500 Cost

Revenue



(B) Algebraic solution:Find x such that

The company breaks even at and 12.530 million cameras.

(C) Graph in a graphing calculator:

x = 2.490

x = 2.490 or 12.530

=

-75.1 ; 22,520.01 -10

x =

-75.1 ; 275.12 - 4(-5)(-156)

2(-5)

-5x2+ 75.1x - 156 = 0

x(94.8 - 5x) = 156 + 19.7x

R(x) = C(x):

0

1

500

15

CostRevenue

0

1

500

15

CostRevenue

(D) Graphical solution:

The company breaks even at and 12.530 million cameras.

(E) Use the results from parts (A) and (B) or (C) and (D):

Profit: 2.490 6 x 6 12.530

Loss: 1 … x 6 2.490 or 12.530 6 x … 15

x = 2.490

0

1

500

15

CostRevenue

MATCHED PROBLEM 4 Use the profit equation from Matched Problem 3:

Profit function

(A) Sketch a graph of the profit function in a rectangular coordinate system.(B) Break-even points occur when Find the break-even points

algebraically to the nearest thousand cameras.

(C) Plot the profit function in an appropriate viewing window.

(D) Find the break-even points graphically to the nearest thousand cameras.

(E) A loss occurs if and a profit occurs if For what outputs (to the nearest thousand cameras) will a loss occur? A profit?

P(x) 7 0.P(x) 6 0,

P(x) = 0.

Domain: 1 … x … 15

= -5x2 + 75.1x - 156

P(x) = R(x) - C(x)



A visual inspection of the plot of a data set might indicate that a parabola wouldbe a better model of the data than a straight line. In that case, rather than using lin-ear regression (Section 1-3) to fit a linear model to the data, we would use quadraticregression on a graphing calculator to find the function of the form that best fits the data.

y = ax2+ bx + c

Outboard Motors Table 1 gives performance data for a boat powered by an Evin-rude outboard motor. Use quadratic regression to find the best model of the form

for fuel consumption y (in miles per gallon) as a function of speedx (in miles per hour). Estimate the fuel consumption (to one decimal place) at a speedof 12 miles per hour.

y = ax2+ bx + c

Enter the data in a graphing calculator (Fig. 5A) and find the quadratic regressionequation (Fig. 5B).The data set and the regression equation are graphed in Figure 5C.Using TRACE we see that the estimated fuel consumption at a speed of 12 mph is 4.5 mpg.

SOLUTION

E X A M P L E 5

TABLE 1

rpm mph mpg

2,500 10.3 4.1

3,000 18.3 5.6

3,500 24.6 6.6

4,000 29.1 6.4

4,500 33.0 6.1

5,000 36.0 5.4

5,400 38.9 4.9

FIGURE 5

MATCHED PROBLEM 5 Refer to Table 1. Use quadratic regression to find the best model of the formfor boat speed y (in miles per hour) as a function of engine speed

x (in revolutions per minute). Estimate the boat speed (in miles per hour, to one dec-imal place) at an engine speed of 3,400 rpm.

� More General Functions: Polynomial and Rational Functions

Linear and quadratic functions are special cases of the more general class of poly-nomial functions.

y = ax2+ bx + c

Polynomial FunctionA polynomial function is a function that can be written in the form

f(x) = anxn+ an - 1x

n - 1+

Á+ a1x + a0

DEFINITION

(A) (B)

0

0

10

50

(C)



Figure 6 shows graphs of representative polynomial functions of degrees 1 through6. The figure suggests some general properties of graphs of polynomial functions.

for n a nonnegative integer, called the degree of the polynomial. The coefficientsare real numbers with . The domain of a polynomial function is

the set of all real numbers.an Z 0a0, a1, . . . , an

(D) F (x) � x2 � 2x � 2 (E) G (x) � 2x4 � 4x2 � x � 1 (F) H(x) � x6 � 7x4 � 14x2 � x � 5

(A) f (x) � x � 2 (B) g (x) � x3 � 2x (C) h(x) � x5 � 5x3 � 4x � 1

x

f (x)

�5

5�5

5

x

g(x)

�5

5�5

5

x

h (x)

�5

5�5

5

x

F (x)

�5

5�5

5

x

G (x)

�5

5�5

5

x

H (x)

�5

5�5

5

FIGURE 6 Graphs of polynomial functions

Notice that the odd-degree polynomial graphs start negative, end positive, andcross the x axis at least once. The even-degree polynomial graphs start positive, endpositive, and may not cross the x axis at all. In all cases in Figure 1, the coefficient ofthe highest-degree term was chosen positive. If any leading coefficient had beenchosen negative, then we would have a similar graph but reflected in the x axis.

A polynomial of degree n can have at most n linear factors. Therefore, the graphof a polynomial function of positive degree n can intersect the x axis at most n times.Note from Figure 6 that a polynomial of degree n may intersect the x axis fewer thann times.

The graph of a polynomial function is continuous, with no holes or breaks.That is,the graph can be drawn without removing a pen from the paper.Also, the graph of apolynomial has no sharp corners. Figure 7 on the next page shows the graphs of twofunctions, one that is not continuous, and the other that is continuous, but with a sharpcorner. Neither function is a polynomial.

Just as rational numbers are defined in terms of quotients of integers, rational func-tions are defined in terms of quotients of polynomials.The following equations specifyrational functions:

p(x) = 3x2- 5x q(x) = 7 r(x) = 0

f(x) =

1x g(x) =

x - 2

x2- x - 6

h(x) =

x3- 8x



Figure 8 shows the graphs of representative rational functions. Note, for example,that in Figure 8A the line is a vertical asymptote for the function.The graph off gets closer to this line as x gets closer to 2.The line in Figure 8A is a horizontalasymptote for the function. The graph of f gets closer to this line as x increases ordecreases without bound.

y = 1x = 2

Discontinuous–breakat x � 0

Continuous, but sharpcorner at (0, �3)

(A) f (x) � (B) h(x) � �x � � 32�x �

x

x

f (x)

x

h (x)

FIGURE 7 Discontinuous and sharp-corner functions

Rational Function

A rational function is any function that can be written in the form

where n(x) and d(x) are polynomials. The domain is the set of all real numberssuch that .d(x) Z 0

f(x) =

n(x)

d(x) d(x) Z 0

DEFINITION

x

f (x)

�5

5�5

5

x

f (x)

�5

5�5

5

x

f (x)

�5

5�5

5

f (x) �x � 3x � 2

f (x) �8

x2 � 4f (x) � x �

1x

(C) (B) (A)

FIGURE 8 Graphs of rational functions

The number of vertical asymptotes of a rational function is atmost equal to the degree of . A rational function has at most one horizontalasymptote (note that the graph in Fig. 8C does not have a horizontal asymptote).Moreover, the graph of a rational function approaches the horizontal asymptote(when one exists) both as x increases and decreases without bound.

d(x)f(x) = n(x)>d(x)

1. (A) (B) x intercepts: y intercept:

(C)

-5-0.7656, 3.2656;Answers to Matched Problems

x

g (x)

10�10

�10 �10

�10

10

10



(D) x intercepts: 3.2656; y intercept: �5

(E) or ; or

(F)

2. (A) .

(B) Vertex: ; maximum: ; range: or

(C) The graph of is the same as the graph of vertically shrunk by a factor of 0.25, reflected in the x axis, and shifted 4 units to theleft and 6 units up.

(D) (E)

g(x) = x2f(x) = -0.25(x + 4)2+ 6

(- q, 6]y … 6f(-4) = 6(-4, 6)

f(x) = -0.25(x + 4)2+ 6

x = -0.3508, 2.8508

(-q, -0.76564 or 33.2656, q)x Ú 3.2656x … -0.7656

-0.7656,

�10

�10

10

10

x

y

�10

�10

10

10

�300

1

300

15

(F) Vertex: ; maximum: ; range: or

3. (A)

(B) ; output of 7.510 million cameraswill produce a maximum profit of 126.001 million dollars.

(C)

(D)

(E)

p(7.510) = $57

P(x) = R(x) - C(x) = -5(x - 7.51)2+ 126.0005

P(x) = R(x) - C(x) = -5x2+ 75.1x - 156

(- q , 6]y … 6f(-4) = 6(-4, 6)

An output of 7.51 million cameras will produce a maximum profit of 126.001 milliondollars. (Notice that maximum profit does not occur at the same output where maxi-mum revenue occurs.)

4. (A) (B) or 12.530 million cameras

(C)

x = 2.490

x

P(x)

�300

0

300

105 15

�300

1

300

15

�300

1

300

15



(D) or 12.530 million cameras

(E) Loss: or ; profit:

5. 22.9 mph

2.490 6 x 6 12.53012.530 6 x … 151 … x 6 2.490

x = 2.490

A In Problems 1–4, complete the square and find the standardform of each quadratic function.

1. 2.

3. 4.

In Problems 5–8, write a brief verbal description of the rela-tionship between the graph of the indicated function (fromProblems 1–4) and the graph of

5. 6.

7. 8.

9. Match each equation with a graph of one of the func-tions f, g, m, or n in the figure.

(A) (B)

(C) (D) y = -(x - 2)2+ 1y = (x + 2)2

- 1

y = (x - 2)2- 1y = -(x + 2)2

+ 1

n(x) = -x2+ 8x - 9m(x) = -x2

+ 6x - 4

g(x) = x2- 2x - 5f(x) = x2

- 4x + 3

y = x2.

n(x) = -x2+ 8x - 9m(x) = -x2

+ 6x - 4

g(x) = x2- 2x - 5f(x) = x2

- 4x + 3

For the functions indicated in Problems 11–14, find each of thefollowing to the nearest integer by referring to the graphs forProblems 9 and 10.

(A) Intercepts (B) Vertex

(C) Maximum or minimum (D) Range

(E) Increasing interval (F) Decreasing interval

11. Function n in the figure for Problem 9

12. Function m in the figure for Problem 10

13. Function f in the figure for Problem 9

14. Function g in the figure for Problem 10

In Problems 15–18, find each of the following:



15.

16.

17.

18.

B In Problems 19–22, write an equation for each graph in theform where a is either 1 or and h andk are integers.

19. 20.

21. 22.

-1y = a(x - h)2 + k,

n(x) = (x - 4)2- 3

m(x) = (x + 1)2- 2

g(x) = -(x + 2)2+ 3

f(x) = -(x - 3)2+ 2

Exercise 2-3

x

y

5�5

f

nm

g

x

y

�5

7�7

5

f

nm

g

x

y

�5

5�5

5Figure for 9

10. Match each equation with a graph of one of the func-tions f, g, m, or n in the figure.

(A) (B)

(C) (D) y = (x + 3)2- 4y = -(x - 3)2

+ 4

y = -(x + 3)2+ 4y = (x - 3)2

- 4

Figure for 10

x

y

�5

5

5

x

y

�5

5�5

5

x

y

�5

5

5



In Problems 23–28, find the vertex form for each quadraticfunction. Then find each of the following:



23.

24.

25.

26.

27.

28.

29. Let Solve each equation graphi-cally to four decimal places.

(A) (B) (C)

30. Let Solve each equationgraphically to four decimal places.

(A) (B) (C)

31. Let Find the maximum value of f tofour decimal places graphically.

32. Let Find the maximum valueof f to four decimal places graphically.

33. Explain under what graphical conditions a quadraticfunction has exactly one real zero.

34. Explain under what graphical conditions a quadraticfunction has no real zeros.

C In Problems 35–38, first write each function in vertex form;then find each of the following (to four decimal places):

(A) Intercepts

(B) Vertex

(C) Maximum or minimum

(D) Range

35.

36.

37.

38. n(x) = -0.15x2 - 0.90x + 3.3

f(x) = -0.12x2 + 0.96x + 1.2

m(x) = 0.20x2 - 1.6x - 1

g(x) = 0.25x2 - 1.5x - 7

f(x) = 100x - 7x2- 10.

f(x) = 125x - 6x2.

g(x) = 8g(x) = 5g(x) = -2

g(x) = -0.6x2+ 3x + 4.

f(x) = -9f(x) = -1f(x) = 4

f(x) = 0.3x2- x - 8.

v(x) = 0.5x2+ 4x + 10

u(x) = 0.5x2- 2x + 5

s(x) = -4x2- 8x - 3

r(x) = -4x2+ 16x - 15

g(x) = x2- 6x + 5

f(x) = x2- 8x + 12

Solve Problems 39–44 graphically to two decimal places usinga graphing calculator.

39.

40.

41.

42.

43.

44.

45. Given that f is a quadratic function with minimumfind the axis, vertex, range, and x

intercepts.

46. Given that f is a quadratic function with maximumfind the axis, vertex, range, and x

intercepts.

In Problems 47–50,

(A) Graph f and g in the same coordinate system.

(B) Solve algebraically to two decimal places.

(C) Solve using parts (A) and (B).

(D) Solve using parts (A) and (B).

47.

48.

49.

50.

51. Give a simple example of a quadratic function that has noreal zeros. Explain how its graph is related to the x axis.

52. Give a simple example of a quadratic function that hasexactly one real zero. Explain how its graph is related tothe x axis.

0 … x … 9 g(x) = 1.1x + 4.8 f(x) = -0.7x2 + 6.3x

0 … x … 8 g(x) = 1.2x + 5.5 f(x) = -0.9x2 + 7.2x

0 … x … 7 g(x) = 0.5x + 3.5 f(x) = -0.7x(x - 7)

0 … x … 10 g(x) = 0.3x + 5 f(x) = -0.4x(x - 10)

f(x) 6 g(x)

f(x) 7 g(x)

f(x) = g(x)

f(x) = f(-3) = -5,

f(x) = f(2) = 4,

1.8x2 - 3.1x - 4.9 7 0

2.8 + 3.1x - 0.9x2 … 0

3.4 + 2.9x - 1.1x2 Ú 0

1.9x2 - 1.5x - 5.6 6 0

7 + 3x - 2x2 = 0

2 - 5x - x2 = 0

Applications

53. Tire mileage. An automobile tire manufacturer collected thedata in the table relating tire pressure x (in pounds per squareinch) and mileage (in thousands of miles):

x Mileage

28 45

30 52

32 55

34 51

36 47

A mathematical model for the data is given by

(A) Complete the following table. Round values of f(x) toone decimal place.

x Mileage f(x)

28 45

30 52

32 55

34 51

36 47

f(x) = -0.518x2 + 33.3x - 481



57. Revenue. The marketing research department for a companythat manufactures and sells memory chips for microcomputersestablished the following price–demand and revenue functions:


where p(x) is the wholesale price in dollars at which x millionchips can be sold, and R(x) is in millions of dollars. Both func-tions have domain

(A) Sketch a graph of the revenue function in a rectangularcoordinate system.

(B) Find the output that will produce the maximum revenue.What is the maximum revenue?

(C) What is the wholesale price per chip (to the nearestdollar) that produces the maximum revenue?

1 … x … 20.

R(x) = xp(x) = x(75 - 3x)

p(x) = 75 - 3x

(B) Sketch the graph of f and the mileage data in the samecoordinate system.

(C) Use values of the modeling function rounded to two deci-mal places to estimate the mileage for a tire pressure of 31pounds per square inch. For 35 pounds per square inch.

(D) Write a brief description of the relationship between tirepressure and mileage.

54. Automobile production. The table shows the retail market shareof passenger cars from Ford Motor Company as a percentageof the U.S. market.

Year Market Share

1980 17.2%

1985 18.8%

1990 20.0%

1995 20.7%

2000 20.2%

2005 17.4%

A mathematical model for this data is given by

where corresponds to 1980.(A) Complete the following table.

x Market Share f(x)

0 17.25 18.8

10 20.015 20.720 20.225 17.4

(B) Sketch the graph of f and the market share data in thesame coordinate system.

(C) Use values of the modeling function f to estimate Ford’smarket share in 2010. In 2015.

(D) Write a brief verbal description of Ford’s market sharefrom 1980 to 2005.

x = 0

f(x) = -0.0206x2+ 0.548x + 16.9

55. Tire mileage. Using quadratic regression on a graphing calcu-lator, show that the quadratic function that best fits the data ontire mileage in Problem 53 is

(coefficients rounded to three significant digits).

56. Automobile production. Using quadratic regression on a graph-ing calculator, show that the quadratic function that best fitsthe data on market share in Problem 54 is

(coefficients rounded to three significant digits).

f(x) = -0.0206x2+ 0.548x + 16.9

f(x) = -0.518x2+ 33.3x - 481

58. Revenue. The marketing research department for a companythat manufactures and sells “notebook” computers establishedthe following price–demand and revenue functions:


where p(x) is the wholesale price in dollars at which x thousandcomputers can be sold, and R(x) is in thousands of dollars. Bothfunctions have domain

(A) Sketch a graph of the revenue function in a rectangularcoordinate system.

(B) Find the output (to the nearest hundred computers) thatwill produce the maximum revenue.What is the maximumrevenue to the nearest thousand dollars?

(C) What is the wholesale price per computer (to the nearestdollar) that produces the maximum revenue?

59. Break-even analysis. Use the revenue function from Problem 57in this exercise and the given cost function:


where x is in millions of chips, and R(x) and C(x) are in mil-lions of dollars. Both functions have domain

(A) Sketch a graph of both functions in the same rectangularcoordinate system.

(B) Find the break-even points to the nearest thousand chips.

(C) For what outputs will a loss occur? A profit?

60. Break-even analysis. Use the revenue function from Problem58, in this exercise and the given cost function:


where x is thousands of computers, and C(x) and R(x) are inthousands of dollars. Both functions have domain 1 … x … 25.

C(x) = 4,000 + 500x

R(x) = x(2,000 - 60x)

1 … x … 20.

C(x) = 125 + 16x

R(x) = x(75 - 3x)

1 … x … 25.

= x(2,000 - 60x)

R(x) = xp(x)

p(x) = 2,000 - 60x


S e c t i o n 2 . 4 Exponential Functions 93

(A) Sketch a graph of both functions in the same rectangularcoordinate system.

(B) Find the break-even points.

(C) For what outputs will a loss occur? Will a profit occur?

61. Profit–loss analysis. Use the revenue and cost functions fromProblem 59 in this exercise:


where x is in millions of chips, and R(x) and C(x) are in mil-lions of dollars. Both functions have domain

(A) Form a profit function P, and graph R, C, and P in thesame rectangular coordinate system.

(B) Discuss the relationship between the intersection pointsof the graphs of R and C and the x intercepts of P.

(C) Find the x intercepts of P to the nearest thousand chips.Find the break-even points to the nearest thousand chips.

(D) Refer to the graph drawn in part (A). Does the maxi-mum profit appear to occur at the same output level asthe maximum revenue? Are the maximum profit and themaximum revenue equal? Explain.

(E) Verify your conclusion in part (D) by finding the output(to the nearest thousand chips) that produces the maxi-mum profit. Find the maximum profit (to the nearestthousand dollars), and compare with Problem 57B.

62. Profit–loss analysis. Use the revenue and cost functions fromProblem 60 in this exercise:


where x is thousands of computers, and R(x) and C(x) are inthousands of dollars. Both functions have domain

(A) Form a profit function P, and graph R, C, and P in thesame rectangular coordinate system.

(B) Discuss the relationship between the intersection pointsof the graphs of R and C and the x intercepts of P.

(C) Find the x intercepts of P to the nearest hundred com-puters. Find the break-even points.

(D) Refer to the graph drawn in part (A). Does the maxi-mum profit appear to occur at the same output level asthe maximum revenue? Are the maximum profit and themaximum revenue equal? Explain.

1 … x … 25.

C(x) = 4,000 + 500x

R(x) = x(2,000 - 60x)

1 … x … 20.

C(x) = 125 + 16x

R(x) = x(75 - 3x)

(E) Verify your conclusion in part (D) by finding the outputthat produces the maximum profit. Find the maximumprofit and compare with Problem 58B.

63. Medicine. The French physician Poiseuille was the first to dis-cover that blood flows faster near the center of an artery thannear the edge. Experimental evidence has shown that the rateof flow (in centimeters per second) at a point x centimetersfrom the center of an artery (see the figure) is given by

Find the distance from the center that the rate of flow is 20centimeters per second. Round answer to two decimal places.

v = f(x) = 1,000(0.04 - x2 ) 0 … x … 0.2

v

Artery

xv

Figure for 63 and 64

64. Medicine. Refer to Problem 63. Find the distance from the cen-ter that the rate of flow is 30 centimeters per second. Roundanswer to two decimal places.

65. The table gives performance data for a boat powered by anEvinrude outboard motor. Find a quadratic regression model

for boat speed y (in miles per hour) as afunction of engine speed (in revolutions per minute). Estimatethe boat speed at 3,100 revolutions per minute.

Table for 65 and 66

rpm mph mpg

1,500 4.5 8.22,000 5.7 6.92,500 7.8 4.83,000 9.6 4.13,500 13.4 3.7

66. The table gives performance data for a boat powered by anEvinrude outboard motor. Find a quadratic regression model

for fuel consumption y (in miles per gal-lon) as a function of engine speed (in revolutions per minute).Estimate the fuel consumption at 2,300 revolutions per minute.

(y = ax2+ bx + c)

(y = ax2+ bx + c)

EXPONENTIAL FUNCTIONS� Exponential Functions� Base e Exponential Functions� Growth and Decay Applications� Compound Interest

This section introduces the important class of functions called exponential functions.These functions are used extensively in modeling and solving a wide variety of real-world problems, including growth of money at compound interest;growth of populationsof people,animals,and bacteria; radioactive decay;and learning associated with the mas-tery of such devices as a new computer or an assembly process in a manufacturing plant.

Section 2-4



� Exponential FunctionsWe start by noting that

are not the same function.Whether a variable appears as an exponent with a constantbase or as a base with a constant exponent makes a big difference. The function g isa quadratic function, which we have already discussed. The function f is a new typeof function called an exponential function. In general,

f(x) = 2x and g(x) = x2

Exponential Function

The equation

defines an exponential function for each different constant b, called the base. Thedomain of f is the set of all real numbers, and the range of f is the set of all positivereal numbers.

b 7 0, b Z 1f(x) = bx

DEFINITION

We require the base b to be positive to avoid imaginary numbers such asWe exclude as a base, since is a con-

stant function, which we have already considered.Asked to hand-sketch graphs of equations such as or , many stu-

dents would not hesitate at all. They would probably makeup tables by assigning integers to x, plot the resulting points, and then join these pointswith a smooth curve as in Figure 1.The only catch is that we have not defined for allreal numbers. From Appendix A, Section A-7, we know what and

mean (that is, , where p is a rational number), but what does

mean? The question is not easy to answer at this time. In fact, a precise definition ofmust wait for more advanced courses, where it is shown that

names a positive real number for x any real number, and that the graph of isindeed as indicated in Figure 1.

It is useful to compare the graphs of and by plotting both on thesame set of coordinate axes, as shown in Figure 2A. The graph of

(Fig. 2B)f(x) = bx b 7 1

y = 2 - xy = 2x

y = 2x

2x

222

222

2p2 - 3.1425, 2 - 3, 22>3, 2-3>5, 21.4,

2x

3Note: 2- x= 1>2x

= (1>2)x.4 y = 2 - xy = 2x

f(x) = 1x= 1b = 1(-2)1>2

= 2-2 = i22.10

5

�5 5x

y

FIGURE 1 y = 2x

y � 2x y � bx

b � 1

y

10

�5 5

5

x

y

Domain � (��, �) Range � (0, �)x

y � (q)x � 2�x y � bx

0 � b � 1

(A) (B)

FIGURE 2 Exponential functions



looks very much like the graph of , and the graph of

(Fig. 2B)

looks very much like the graph of Note that in both cases the x axis is a hor-izontal asymptote for the graphs.

The graphs in Figure 2 suggest the following important general properties ofexponential functions, which we state without proof:

y = 2 - x.

f(x) = bx 0 6 b 6 1

y = 2x

BASIC PROPERTIES OF THE GRAPH OF

1. All graphs will pass through the point (0, 1). bo = 1 for any permissible base b.

2. All graphs are continuous curves, with no holes or jumps.3. The x axis is a horizontal asymptote.4. If , then increases as x increases.5. If , then decreases as x increases.bx0 6 b 6 1

bxb 7 1

f(x) � bx, b>0, b � 1THEOREM 1

Recall that the graph of a rational function has at most one horizontal asymptote and thatit approaches the horizontal asymptote (if one exists) both as and as (see Section 2-3).The graph of an exponential function, on the other hand, approaches itshorizontal asymptote as or as but not both. In particular, there is no ra-tional function that has the same graph as an exponential function.

x : -q,x : q

x : - qx : q

I N S I G H T

Graphing Exponential Functions Sketch a graph of y = (12)4x, -2 … x … 2.E X A M P L E 1

The use of a calculator with the key , or its equivalent, makes the graphingof exponential functions almost routine. Example 1 illustrates the process.

yx

SOLUTION Use a calculator to create the table of values shown. Plot these points, and then jointhem with a smooth curve as in Figure 3.

x y

0.031

0.125

0 0.50

1 2.00

2 8.00

-1

-2

10

�2 2

5

y

x

FIGURE 3 Graph of y = (12)4x

MATCHED PROBLEM 1 Sketch a graph of y = (12)4 - x, -2 … x … 2.



Graph the functions and on the same set of coordinate axes.At which values of x do the graphs intersect? For which values of x is the graphof f above the graph of g? Below the graph of g? Are the graphs close togetheras x increases without bound? Are the graphs close together as x decreaseswithout bound? Discuss.

Exponential functions, whose domains include irrational numbers, obey the familiarlaws of exponents discussed in Appendix A, Section A-6 for rational exponents. Wesummarize these exponent laws here and add two other important and usefulproperties.

g(x) = 3xf(x) = 2xExplore & Discuss 1

PROPERTIES OF EXPONENTIAL FUNCTIONSFor a and b positive, and x and y real,

1. Exponent laws:

2. if and only if

3. For

if and only if If a5= 25, then a = 2.a = bax

= bx

x Z 0,

5t + 1 = 3t - 3, and t = -2.If 75t + 1

= 73t - 3, thenx = yax= ay

(ax)y= axy (ab)x

= axbx a a

bbx

=

ax

bx

= 4-3y= 42y - 5y42y

45yaxay= ax + y

ax

ay = ax - y

a Z 1, b Z 1,THEOREM 2

� Base e Exponential FunctionsOf all the possible bases b we can use for the exponential function whichones are the most useful? If you look at the keys on a calculator, you will probablysee and . It is clear why base 10 would be important, because our numbersystem is a base 10 system. But what is e, and why is it included as a base? It turns outthat base e is used more frequently than all other bases combined. The reason forthis is that certain formulas and the results of certain processes found in calculus andmore advanced mathematics take on their simplest form if this base is used. This iswhy you will see e used extensively in expressions and formulas that model real-worldphenomena. In fact, its use is so prevalent that you will often hear people refer to

as the exponential function.The base e is an irrational number, and like �, it cannot be represented exactly by

any finite decimal fraction. However, e can be approximated as closely as we like byevaluating the expression

(1)

for sufficiently large x. What happens to the value of expression (1) as x increaseswithout bound? Think about this for a moment before proceeding. Maybe you guessedthat the value approaches 1, because

approaches 1, and 1 raised to any power is 1. Let us see if this reasoning is correctby actually calculating the value of the expression for larger and larger values of x.Table 1 summarizes the results.

1 +

1x

a1 +

1xbx

y = ex

ex10x

y = bx,



Interestingly, the value of expression (1) is never close to 1, but seems to beapproaching a number close to 2.7183. In fact, as x increases without bound, the valueof expression (1) approaches an irrational number that we call e. The irrationalnumber e to 12 decimal places is

Compare this value of e with the value of from a calculator. Exactly who discov-ered the constant e is still being debated. It is named after the great Swiss mathe-matician Leonhard Euler (1707–1783).

e1

e � 2.718 281 828 459

TABLE 1

1 2

10 2.59374…

100 2.70481…

1,000 2.71692…

10,000 2.71814…

100,000 2.71827…

1,000,000 2.71828…

x a1 +

1xbx

Exponential Function with Base eExponential functions with base e and base respectively, are defined by

and

Domain:

Range: (0, q)

1- q, q)

y = e - xy = ex

1>e,

DEFINITION

10

�5 5

5

y � e�x y � ex

x

y

Graph the functions and on the same set of co-ordinate axes. At which values of x do the graphs intersect? For positive valuesof x, which of the three graphs lies above the other two? Below the other two?How does your answer change for negative values of x?

� Growth and Decay ApplicationsFunctions of the form where c and k are constants and the independentvariable t represents time, are often used to model population growth and radioac-tive decay. Note that if then So the constant c represents the initialpopulation (or initial amount). The constant k is called the relative growth rate and

y = c.t = 0,

y = cekt,

h(x) = 3xf(x) = ex, g(x) = 2x,Explore & Discuss 2



has the following interpretation: Suppose that models the population growthof a country, where y is the number of persons and t is time in years. If the relativegrowth rate is then at any time t, the population is growing at a rate of persons (that is, 2% of the population) per year.

We say that population is growing continuously at relative growth rate k to meanthat the population y is given by the model y = cekt.

0.02yk = 0.02,

y = cekt

E X A M P L E 2 Exponential Growth Cholera, an intestinal disease, is caused by a cholera bac-terium that multiplies exponentially by cell division. The number of bacteria growscontinuously at relative growth rate 1.386, that is,

where N is the number of bacteria present after t hours and is the number of bac-teria present at the start . If we start with 25 bacteria, how many bacteria (to the nearest unit) will be present:

(A) In 0.6 hour? (B) In 3.5 hours?

(t = 0)N0

N = N0e1.386t

SOLUTION Substituting into the preceding equation, we obtain

The graph is shown in Figure 4.N = 25e1.386t

N0 = 25

(A) Solve for N when

Use a calculator.

(B) Solve for N when

Use a calculator. = 3,197 bacteria

N = 25e1.386(3.5)

t = 3.5:

= 57 bacteriaN = 25e1.386(0.6)

t = 0.6:

Refer to the exponential growth model for cholera in Example 2. If we start with 55bacteria, how many bacteria (to the nearest unit) will be present

(A) In 0.85 hour? (B) In 7.25 hours?

MATCHED PROBLEM 2

E X A M P L E 3 Exponential Decay Cosmic-ray bombardment of the atmosphere produces neutrons,which in turn react with nitrogen to produce radioactive carbon-14 . Radioactive

enters all living tissues through carbon dioxide, which is first absorbed by plants.14C(14C)

10,000

50

Time (hours)

t

N

FIGURE 4



As long as a plant or animal is alive, is maintained in the living organism at a con-stant level. Once the organism dies, however, decays according to the equation

where A is the amount present after t years and is the amount present at timeIf 500 milligrams of is present in a sample from a skull at the time of death,

how many milligrams will be present in the sample in

(A) 15,000 years? (B) 45,000 years?

Compute answers to two decimal places.

14Ct = 0.A0

A = A0e- 0.000124t

14C

14C

100

200

300

400

500

50,0000

Mill

igra

ms

Years

A

t

SOLUTION Substituting in the decay equation, we have

See the graph in Figure 5.A = 500e - 0.000124t

A0 = 500

(A) Solve for A when

Use a calculator.

(B) Solve for A when

Use a calculator. = 1.89 milligrams

A = 500e - 0.000124(45,000)

t = 45,000:

= 77.84 milligrams

A = 500e - 0.000124(15,000)

t = 15,000:

Refer to the exponential decay model in Example 3. How many milligrams of would have to be present at the beginning in order to have 25 milligrams presentafter 18,000 years? Compute the answer to the nearest milligram.

14CMATCHED PROBLEM 3

Explore & Discuss 3 (A) On the same set of coordinate axes, graph the three decay equationsfor

(B) Identify any asymptotes for the three graphs in part (A).

(C) Discuss the long-term behavior for the equations in part (A).

If you buy a new car, it is likely to depreciate in value by several thousand dollarsduring the first year you own it.You would expect the value of the car to decrease ineach subsequent year, but not by as much as in the previous year. If you drive the carlong enough, its resale value will get close to zero. An exponential decay functionwill often be a good model of depreciation; a linear or quadratic function would notbe suitable (why?). We can use exponential regression on a graphing calculator tofind the function of the form that best fits a data set.y = abx

A0 = 10, 20, and 30.A = A0e-0.35t, t Ú 0,

FIGURE 5



(A) (B)0

0

20000

11

(C)

TABLE 2

x Value ($)

1 12,575

2 9,455

3 8,115

4 6,845

5 5,225

6 4,485

TABLE 3

x Value ($)

1 23,125

2 19,050

3 15,625

4 11,875

5 9,450

6 7,125

Enter the data into a graphing calculator (Fig. 6A) and find the exponential regres-sion equation (Fig. 6B). The estimated purchase price is The dataset and the regression equation are graphed in Figure 6C. Using TRACE, we see that theestimated value after 10 years is $1,959.

y1(0) = $14,910.SOLUTION

Table 3 gives the market value of a luxury sedan (in dollars) x years after its pur-chase. Find an exponential regression model of the form for this data set.Estimate the purchase price of the sedan. Estimate the value of the sedan 10 yearsafter its purchase. Round answers to the nearest dollar.

y = abxMATCHED PROBLEM 4

� Compound InterestThe fee paid to use another’s money is called interest. It is usually computed as apercent (called interest rate) of the principal over a given period of time. If, at the endof a payment period, the interest due is reinvested at the same rate, then the interestearned as well as the principal will earn interest during the next payment period.Interest paid on interest reinvested is called compound interest, and may be calcu-lated using the following compound interest formula:

E X A M P L E 4 Depreciation Table 2 gives the market value of a minivan (in dollars) x years afterits purchase. Find an exponential regression model of the form for this dataset. Estimate the purchase price of the van. Estimate the value of the van 10 years afterits purchase. Round answers to the nearest dollar.

y = abx

FIGURE 6



If a principal P (present value) is invested at an annual rate r (expressed as adecimal) compounded m times a year, then the amount A (future value) in theaccount at the end of t years is given by

Compound interest formula

Note: P could be replaced by but convention dictates otherwise. For given rand m, the amount A is equal to the principal P multiplied by the exponentialfunction , where b = (1 + r>m)m.bt

A0,

A = Pa1 +

rmbmt

E X A M P L E 5 Compound Growth If $1,000 is invested in an account paying 10% compoundedmonthly, how much will be in the account at the end of 10 years? Compute the answerto the nearest cent.

SOLUTION We use the compound interest formula as follows:

Use a calculator.

The graph of

for is shown in Figure 7.0 … t … 20

A = 1,000a1 +

0.1012b12t

= $2,707.04

= 1,000a1 +

0.1012b (12)(10)

A = Pa1 +

rmbmt

MATCHED PROBLEM 5

t

$10,000

$5,000

20100

Years

A

FIGURE 7

If you deposit $5,000 in an account paying 9% compounded daily, how much willyou have in the account in 5 years? Compute the answer to the nearest cent.

Explore & Discuss 4 Suppose that $1,000 is deposited in a savings account at an annual rate of 5%.Guess the amount in the account at the end of 1 year if interest is compounded(1) quarterly, (2) monthly, (3) daily, (4) hourly. Use the compound interest formulato compute the amounts at the end of 1 year to the nearest cent. Discuss theaccuracy of your initial guesses.

The formula for compound interest is summarized on the next page for conve-nient reference.



1. 2. (A) 179 bacteria (B) 1,271,659 bacteria

COMPOUND INTEREST FORMULA

where

amount (future value) at the end of t years

principal (present value)

annual rate, expressed as a decimal

number of compounding periods per year

time in yearst =

m =

r =

P =

A =

A = Pa1 + rmbmt

SUMMARY

Answers to Matched Problems

x

y

5

2�2

10

3. 233 mg 4. Purchase price: $30,363;value after 10 yr: $2,864

5. $7,841.13

6. (A) $9,155.23 (B) $9,156.29

A1. Match each equation with the graph of f, g, h, or k in the

figure.

(A) (B)

(C) (D) y = (13)xy = 4x

y = (0.2)xy = 2x

Exercise 2-4

x

y

5

�5 5

f g h k

x

y

5

�5 5

f g h k

2. Match each equation with the graph of f, g, h, or k in thefigure.

(A) (B)

(C) (D) y = 3xy = 5x

y = (0.5)xy = (14)x



x

y

�5

5�5

5

Graph each function in Problems 3–14 over the indicated interval.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

Simplify each expression in Problems 15–20.

15. 16.

17. 18.

19. 20.

B In Problems 21–28, describe verbally the transformations thatcan be used to obtain the graph of g from the graph of f (seeSection 2-2).

21.

22.

23.

24.

25.

26.

27.

28.

29. Use the graph of f shown in the figure to sketch thegraph of each of the following.

(A) (B)

(C) (D)

30. Use the graph of f shown in the figure to sketch thegraph of each of the following.

(A) (B)

(C) (D) y = 4 - f(x + 2)y = 2f(x) - 4

y = f(x - 3)y = f(x) + 2

y = 2 - f(x - 3)y = 3f(x) - 2

y = f(x + 2)y = f(x) - 1

g(x) = 0.5e-(x - 1); f(x) = e-x

g(x) = 2e-(x + 2); f(x) = e-x

g(x) = ex- 2; f(x) = ex

g(x) = ex+ 1; f(x) = ex

g(x) = -3x; f(x) = 3x

g(x) = 3x + 1; f(x) = 3x

g(x) = 2x - 2; f(x) = 2x

g(x) = -2x; f(x) = 2x

(3e-1.4x)2(2e1.2t)3

ex

e1 - x

ex - 3

ex - 4

103x - 1104 - x(43x)2y

f(t) = 100e-0.1t; [-5, 5]g(t) = 10e-0.2t; [-5, 5]

y = 10e0.2x; [-10, 10]y = 100e0.1x; [-5, 5]

y = -ex; [-3, 3]y = -e-x; [-3, 3]

g(x) = -3-x; [-3, 3]f(x) = -5x; [-2, 2]

y = (13)x

= 3-x; [-3, 3]y = (15)x

= 5-x; [-2, 2]

y = 3x; [-3, 3]y = 5x; [-2, 2]

In Problems 31–40, graph each function over the indicatedinterval.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41. Find all real numbers a such that Explain why this does not violate the second exponentialfunction property in Theorem 2 on page 96.

42. Find real numbers a and b such that butExplain why this does not violate the

third exponential function property in Theorem 2 on page 96.

Solve each equation in Problems 43–48 for x.

43. 44.

45. 46.

47. 48.

C Solve each equation in Problems 49–52 for x. (Remember:and )

49.

50.

51.

52.

Graph each function in Problems 53–56 over the indicated interval.

53. 54.

55. 56.

In Problems 57–60, approximate the real zeros of each functionto two decimal places.

57. 58.

59. 60. f(x) = 7 - 2x2+ 2-xf(x) = 2 + 3x + 10x

f(x) = 5 - 3-xf(x) = 4x- 7

N =

2001 + 3e-t; [0, 5]N =

1001 + e-t; [0, 5]

m(x) = x(3-x); [0, 3]h(x) = x(2x); [-5, 0]

x2ex- 5xex

= 0

3xe-x+ x2e-x

= 0

2xe-x= 0

(x - 3)ex= 0

e-xZ 0.ex

Z 0

(1 - x)5= (2x - 1)553

= (x + 2)3

7x2= 72x + 345x - x2

= 4-6

53x= 54x - 2102 - 3x

= 105x - 6

a4= b4.

a Z b

a2= a-2.

y = 2-x2; [-3, 3]

y = e-x2; [-3, 3]

M(x) = ex>2+ e-x>2; [-5, 5]

C(x) =

ex+ e-x

2; [-5, 5]

y = e-ƒx ƒ; [-3, 3]

y = e ƒx ƒ; [-3, 3]

y = 2 + ex - 2; [-1, 5]

y = -3 + e1 + x; [-4, 2]

G(t) = 3t>100; [-200, 200]

f(t) = 2t>10; [-30, 30]

Figure for 29 and 30



Applications

In all problems involving days, a 365-day year is assumed.

61. Finance. Suppose that $2,500 is invested at 7% compoundedquarterly. How much money will be in the account in

(A) year? (B) 15 years?

Compute answers to the nearest cent.

62. Finance. Suppose that $4,000 is invested at 6% compoundedweekly. How much money will be in the account in

(A) year? (B) 10 years?

Compute answers to the nearest cent.

63. Finance. A person wishes to have $15,000 cash for a new car5 years from now. How much should be placed in an accountnow, if the account pays 6.75% compounded weekly? Computethe answer to the nearest dollar.

64. Finance. A couple just had a baby. How much should they in-vest now at 5.5% compounded daily in order to have $40,000 forthe child’s education 17 years from now? Compute the answerto the nearest dollar.

65. Money growth. BanxQuote operates a network of Web sitesproviding real-time market data from leading financialproviders. The following rates for 12-month certificates ofdeposit were taken from the Web sites:

(A) Stonebridge Bank, 5.40% compounded monthly

(B) DeepGreen Bank, 4.95% compounded daily

(C) Provident Bank, 5.15% compounded quarterly

Compute the value of $10,000 invested in each account at theend of 1 year.

66. Money growth. Refer to Problem 65. The following rates for 60-month certificates of deposit were also taken from Banx-Quote Web sites:

(A) Oriental Bank & Trust, 5.50% compounded quarterly

(B) BMW Bank of North America, 5.12% compoundedmonthly

(C) BankFirst Corporation, 4.86% compounded daily

Compute the value of $10,000 invested in each account at theend of 5 years.

67. Advertising. A company is trying to introduce a new productto as many people as possible through television advertising ina large metropolitan area with 2 million possible viewers. Amodel for the number of people N (in millions) who are awareof the product after t days of advertising was found to be

Graph this function for What value does N tendto as t increases without bound?

68. Learning curve. People assigned to assemble circuit boardsfor a computer manufacturing company undergo on-the-jobtraining. From past experience it was found that the learningcurve for the average employee is given by

where N is the number of boards assembled per day after t daysof training. Graph this function for What is the0 … t … 30.

N = 40(1 - e-0.12t)

0 … t … 50.

N = 2(1 - e-0.037t)

12

34

maximum number of boards an average employee can be ex-pected to produce in 1 day?

69. Sports salaries. Table 4 shows the average salaries for players inMajor League Baseball (MLB) and the National BasketballAssociation (NBA) in selected years since 1990.(A) Let x represent the number of years since 1990 and find

an exponential regression model for the aver-age salary in MLB. Use the model to estimate the aver-age salary (to the nearest thousand dollars) in 2015.

(B) The average salary in MLB in 2000 was 1.984 million.How does this compare with the value given by themodel of part (A)? How would the inclusion of the year2000 data affect the estimated average salary in 2015?Explain.

(y = abx)

TABLE 4 Average Salary (thousand $)

Year MLB NBA

1990 589 750

1993 1,062 1,300

1996 1,101 2,000

1999 1,724 2,400

2002 2,300 4,500

2005 2,633 5,000

70. Sports salaries. Refer to Table 4.(A) Let x represent the number of years since 1990 and find

an exponential regression model for the aver-age salary in the NBA. Use the model to estimate the av-erage salary (to the nearest thousand dollars) in 2015.

(B) The average salary in the NBA in 1997 was $2.2 million.How does this compare with the value given by themodel of part (A)? How would the inclusion of the year1997 data affect the estimated average salary in 2015? Explain.

71. Marine biology. Marine life is dependent upon the microscopicplant life that exists in the photic zone, a zone that goes to adepth where about 1% of the surface light remains. In somewaters with a great deal of sediment, the photic zone may godown only 15 to 20 feet. In some murky harbors, the intensityof light d feet below the surface is given approximately by

I = I0e-0.23d

(y = abx)


S e c t i o n 2 . 4 Logarithmic Functions 105

What percentage of the surface light will reach a depth of

(A) 10 feet? (B) 20 feet?

72. Marine biology. Refer to Problem 71. Light intensity I relative todepth d (in feet) for one of the clearest bodies of water in the world,the Sargasso Sea in the West Indies, can be approximated by

where is the intensity of light at the surface.What percentageof the surface light will reach a depth of

(A) 50 feet? (B) 100 feet?

73. HIV/AIDS epidemic. The Joint United Nations Program onHIV/AIDS reported that HIV had infected 65 million peopleworldwide prior to 2006. Assume that number increases con-tinuously at a relative growth rate of 2%.(A) Write an equation that models the worldwide spread of

HIV, letting 2006 be year 0.

(B) Based on the model, how many people (to the nearestmillion) had been infected prior to 2002? How manywould be infected prior to 2014?

(C) Sketch a graph of this growth equation from 2006 to 2014.

74. HIV/AIDS epidemic. The Joint United Nations Program onHIV/AIDS reported that 25 million people worldwide had diedof AIDS prior to 2006. Assume that number increases contin-uously at a relative growth rate of 1%.(A) Write an equation that models total worldwide deaths

from AIDS, letting 2006 be year 0.

(B) Based on the model, how many people (to the nearestmillion) had died from AIDS prior to 2002? How manywould die from AIDS prior to 2014?

(C) Sketch a graph of this growth equation from 2006 to 2014.

75. World population growth. It took from the dawn of humanity to1830 for the population to grow to the first billion people, just100 more years (by 1930) for the second billion, and 3 billionmore were added in only 60 more years (by 1990). In 2006, theestimated world population was 6.5 billion with a relativegrowth rate of 1.14%.(A) Write an equation that models the world population

growth, letting 2006 be year 0.

(B) Based on the model, what is the expected worldpopulation (to the nearest hundred million) in 2015? In 2030?

(C) Sketch a graph of the equation found in part (A). Coverthe years from 2006 through 2030.

I0

I = I0e-0.00942d

76. Population growth in Ethiopia. In 2006, the estimated popula-tion in Ethiopia was 75 million people with a relative growthrate of 2.3%.

(A) Write an equation that models the population growth inEthiopia, letting 2006 be year 0.

(B) Based on the model, what is the expected populationin Ethiopia (to the nearest million) in 2015? In 2030?

(C) Sketch a graph of the equation found in part (A). Coverthe years from 2006 through 2030.

77. Internet growth. The number of Internet hosts grew very rapidlyfrom 1994 to 2006 (Table 5).

(A) Let x represent the number of years since 1994. Find anexponential regression model for this data setand estimate the number of hosts in 2015 (to the nearestmillion).

(B) Discuss the implications of this model if the number ofInternet hosts continues to grow at this rate.

(y = abx)

TABLE 6

Year of LifeBirth Expectancy

1970 70.8

1975 72.6

1980 73.7

1985 74.7

1990 75.4

1995 75.9

2000 76.9

2005 77.7

TABLE 5 Internet Hosts (Millions)

Year Hosts

1994 2.4

1997 16.1

2000 72.4

2003 171.6

2006 394.0

78. Life expectancy. Table 6 shows the life expectancy (in years) atbirth for residents of the United States from 1970 to 2005. Letx represent years since 1970. Find an exponential regressionmodel for this data and use it to estimate the life expectancy fora person born in 2015.

LOGARITHMIC FUNCTIONS� Inverse Functions� Logarithmic Functions� Properties of Logarithmic Functions� Calculator Evaluation of Logarithms� Applications

Find the exponential function keys and on your calculator. Close to thesekeys you will find and keys. The latter represent logarithmic functions, andeach is closely related to the exponential function it is near. In fact, the exponential

LNLOG

ex10x

Section 2-5

Source: Internet Software Consortium



x

f (x)

�5

5�5

5

�5

5�5

5

(A) f (x) � (B) g(x) � x2

�x �2

g(x)

x

FIGURE 1

Because both f and g are functions, each domain value corresponds to exactly onerange value. For which function does each range value correspond to exactly onedomain value? This is the case only for function f. Note that for the range value 2, thecorresponding domain value is 4. For function g the range value 2 corresponds toboth and 4. Function f is said to be one-to-one. In general,-4

One-to-One FunctionsA function f is said to be one-to-one if each range value corresponds to exactly onedomain value.

DEFINITION

It can be shown that any continuous function that is either increasing or decreas-ing for all domain values is one-to-one. If a continuous function increases for somedomain values and decreases for others, it cannot be one-to-one. Figure 1 shows anexample of each case.

Explore & Discuss 1 Graph and For a range value of 4, what are thecorresponding domain values for each function? Which of the two functions isone-to-one? Explain why.

g(x) = x2.f(x) = 2x

function and the corresponding logarithmic function are said to be inverses of eachother. In this section we will develop the concept of inverse functions and use it to de-fine a logarithmic function as the inverse of an exponential function.We will then in-vestigate basic properties of logarithmic functions, use a calculator to evaluate themfor particular values of x, and apply them to real-world problems.

Logarithmic functions are used in modeling and solving many types of problems.For example, the decibel scale is a logarithmic scale used to measure sound intensity,and the Richter scale is a logarithmic scale used to measure the strength of the forceof an earthquake. An important business application has to do with finding the timeit takes money to double if it is invested at a certain rate compounded a given numberof times a year or compounded continuously. This requires the solution of anexponential equation, and logarithms play a central role in the process.

� Inverse Functions

Look at the graphs of and in Figure 1:g(x) =

ƒ x ƒ

2f(x) =

x

2



Inverse of a FunctionIf f is a one-to-one function, then the inverse of f is the function formed by inter-changing the independent and dependent variables for f.Thus, if (a, b) is a point onthe graph of f, then (b, a) is a point on the graph of the inverse of f.

Note: If f is not one-to-one, then f does not have an inverse.

DEFINITION

A number of important functions in any library of elementary functions are theinverses of other basic functions in the library. In this course, we are interested in theinverses of exponential functions, called logarithmic functions.

� Logarithmic FunctionsIf we start with the exponential function f defined by

(1)

and interchange the variables, we obtain the inverse of f:

(2)

We call the inverse the logarithmic function with base 2, and write

We can graph by graphing since they are equivalent.Any orderedpair of numbers on the graph of the exponential function will be on the graph of thelogarithmic function if we interchange the order of the components. For example,(3, 8) satisfies equation (1) and (8, 3) satisfies equation (2).The graphs of and

are shown in Figure 2. Note that if we fold the paper along the dashedline in Figure 2, the two graphs match exactly. The line is a line of sym-metry for the two graphs.

y = xy = xy = log2 x

y = 2x

x = 2y,y = log2 x

y = log2 x if and only if x = 2y

x = 2y

y = 2x

Exponential LogarithmicFunction Function

x yx = 2yy = 2x

0 1 1 0

1 2 2 1

2 4 4 2

3 8 8 3

-112

12-1

-214

14-2

-318

18-3

5�5

�5

5

10

10

y � 2x y � xy

x

x � 2y

ory � log2 x

FIGURE 2

C Orderedpairs

reversedSq q

Starting with a one-to-one function f, we can obtain a new function called theinverse of f as follows:

In general, since the graphs of all exponential functions of the formare either increasing or decreasing (see Section 2-2),

exponential functions have inverses.f(x) = bx, b Z 1, b 7 0,



(A) is equivalent to

(B) is equivalent to

(C) is equivalent to log2(18) = -31

8 = 2-3

log36 6 =126 = 236

log4 64 = 364 = 43

Logarithmic FunctionsThe inverse of an exponential function is called a logarithmic function. For and

Logarithmic form Exponential form

The log to the base b of x is the exponent to which b must be raised to obtain x.[Remember: A logarithm is an exponent.] The domain of the logarithmic function isthe set of all positive real numbers, which is also the range of the correspondingexponential function; and the range of the logarithmic function is the set of all realnumbers, which is also the domain of the corresponding exponential function.Typicalgraphs of an exponential function and its inverse, a logarithmic function, are shownin the figure in the margin.

y = logb x is equivalent to x = by

b Z 1,b 7 0

DEFINITION

Because the domain of a logarithmic function consists of the positive real numbers, theentire graph of a logarithmic function lies to the right of the y axis. In contrast, the graphsof polynomial and exponential functions intersect every vertical line, and the graphs ofrational functions intersect all but a finite number of vertical lines.

I N S I G H T

The following examples involve converting logarithmic forms to equivalentexponential forms, and vice versa.

E X A M P L E 1 Logarithmic–Exponential Conversions Change each logarithmic form to anequivalent exponential form:

(A) (B) (C) log2(14) = -2log9 3 =

12log5 25 = 2

SOLUTION (A) is equivalent to

(B) is equivalent to

(C) is equivalent to 14 = 2-2log2(

14) = -2

3 = 91/2log9 3 =12

25 = 52log5 25 = 2

MATCHED PROBLEM 1 Change each logarithmic form to an equivalent exponential form:

(A) (B) (C) log3(19) = -2log4 2 =

12log3 9 = 2

Exponential–Logarithmic Conversions Change each exponential form to anequivalent logarithmic form:

(A) (B) (C) 18 = 2-36 = 23664 = 43

E X A M P L E 2

SOLUTION

x

y

5�5

5

10

10

�5

y � bx

y � x

y � logb x



MATCHED PROBLEM 2 Change each exponential form to an equivalent logarithmic form:

(A) (B) (C)

To gain a little deeper understanding of logarithmic functions and their relation-ship to the exponential functions, we consider a few problems where we want to findx, b, or y in given the other two values. All values are chosen so that theproblems can be solved exactly without a calculator.

y = logb x,

13 = 3-13 = 2949 = 72

E X A M P L E 3 Solutions of the Equation Find y, b, or x, as indicated.

(A) Find y: (B) Find x:

(C) Find y: (D) Find b: logb 100 = 2y = log8 4

log2 x = -3y = log4 16

y = logb x

SOLUTION (A) is equivalent to Thus,

(B) is equivalent to Thus,

(C) is equivalent to

Thus,

(D) is equivalent to Thus,

Recall that b cannot be negative.b = 10

100 = b2.logb 100 = 2

y =23

3y = 2

4 = 8y or 22= 23y

y = log8 4x =

1

23 =

18

x = 2-3.log2 x = -3y = 2

16 = 4y.y = log4 16

MATCHED PROBLEM 3 Find y, b, or x, as indicated.

(A) Find (B) Find

(C) Find

� Properties of Logarithmic FunctionsLogarithmic functions have many powerful and useful properties.We list eight basicproperties in Theorem 1.

b: logb 1,000 = 3

x: log3 x = -1y: y = log9 27

PROPERTIES OF LOGARITHMIC FUNCTIONSIf b, M, and N are positive real numbers, and p and x are real numbers, then

1. 5.

2. 6.

3. 7.

4. 8. if and only if M = Nlogb M = logb Nblogb x= x, x 7 0

logb Mp= p logb Mlogb bx

= x

logb M

N= logb M - logb Nlogb b = 1

logb MN = logb M + logb Nlogb 1 = 0

b Z 1,THEOREM 1

The first four properties in Theorem 1 follow directly from the definition of a log-arithmic function. Here we will sketch a proof of property 5. The other propertiesare established in a similar way. Let

u = logb M and v = logb N



Or, in equivalent exponential form,

Now, see if you can provide reasons for each of the following steps:

logb MN = logb bubv= logb bu + v

= u + v = logb M + logb N

M = bu and N = bv

Using Logarithmic Properties

(A)

(B)

(C)

(D)loge x

loge b=

loge(blogb x)

loge b=

(logb x)(loge b)

loge b= logb x

ex loge b= eloge b

x

= bx

=35(logb w + logb x)=

35 logb wxlogb(wx)3/5

= logb w + logb x - logb y - logb z

= logb wx - logb yz

= logb w + logb x - (logb y + logb z)logb

wxyz

E X A M P L E 4

MATCHED PROBLEM 4 Write in simpler forms, as in Example 4.

(A) (B) (C) (D)log2 x

log2 b2u log2 blogbaR

Sb2/3

logb R

ST

The following examples and problems, although somewhat artificial, will give youadditional practice in using basic logarithmic properties.

Solving Logarithmic Equations Find x so that

32 logb 4 -

23 logb 8 + logb 2 = logb x

Property 7

Properties 5 and 6

Property 8 x = 4

logb 4 = logb x

logb 8 # 2

4= logb x

logb 8 - logb 4 + logb 2 = logb x

logb 43/2- logb 82/3

+ logb 2 = logb x

32 logb 4 -23 logb 8 + logb 2 = logb xSOLUTION

E X A M P L E 5

MATCHED PROBLEM 5 Find x so that 3 logb 2 +12 logb 25 - logb 20 = logb x.



� Calculator Evaluation of LogarithmsOf all possible logarithmic bases, the base e and the base 10 are used almost exclu-sively. Before we can use logarithms in certain practical problems, we need to be ableto approximate the logarithm of any positive number either to base 10 or to base e.And conversely, if we are given the logarithm of a number to base 10 or base e, weneed to be able to approximate the number. Historically, tables were used for thispurpose, but now calculators make computations faster and far more accurate.

Common logarithms (also called Briggsian logarithms) are logarithms with base10. Natural logarithms (also called Napierian logarithms) are logarithms with base e.Most calculators have a key labeled “log” (or “LOG”) and a key labeled “ln” (or“LN”).The former represents a common (base 10) logarithm and the latter a natural(base e) logarithm. In fact, “log” and “ln” are both used extensively in mathematicalliterature, and whenever you see either used in this book without a base indicated, theywill be interpreted as follows:

Common logarithm:Natural logarithm:

Finding the common or natural logarithm using a calculator is very easy. On somecalculators, you simply enter a number from the domain of the function and press or . On other calculators, you press either or , enter a number from the do-main, and then press . Check the user’s manual for your calculator.ENTER

LNLOGLN

LOG

ln x means loge xlog x means log10 x

Solving Logarithmic Equations Solve: log10 x + log10(x + 1) = log10 6.

Property 5Property 8Solve by factoring.

We must exclude since the domain of the function is or hence, is the only solution.x = 2(-1, q );

x 7 -1log10(x + 1)x = -3,

x = -3, 2

(x + 3)(x - 2) = 0

x2+ x - 6 = 0

x(x + 1) = 6

log10[x(x + 1)] = log10 6

log10 x + log10(x + 1) = log10 6SOLUTION

E X A M P L E 6

MATCHED PROBLEM 6 Solve: log3 x + log3(x - 3) = log3 10.

Discuss the relationship between each of the following pairs of expressions. Ifthe two expressions are equivalent, explain why. If they are not, give an example.

(A)

(B)

(C)

(D) logb M + logb N; logb(M + N)

logb M + logb N; logb MN

logb M - logb N; logb M

N

logb M - logb N; logb M

logb N

Explore & Discuss 2

Calculator Evaluation of Logarithms Use a calculator to evaluate each to sixdecimal places:

(A) log 3,184 (B) ln 0.000 349 (C) log(-3.24)

E X A M P L E 7



We now turn to the second problem mentioned previously: Given the logarithmof a number, find the number.We make direct use of the logarithmic– exponential re-lationships, which follow from the definition of logarithmic function given at the be-ginning of this section.

ln x � y is equivalent to x � ey

log x � y is equivalent to x � 10 y

(A)

(B)

(C) * �3.24 is not in the domain of the log function.log(-3.24) = Error

ln 0.000 349 = -7.960 439

log 3,184 = 3.502 973SOLUTION

* Some calculators use a more advanced definition of logarithms involving complex numbers and will dis-play an ordered pair of real numbers as the value of You should interpret such a result as anindication that the number entered is not in the domain of the logarithm function as we have defined it.

log(-3.24).

MATCHED PROBLEM 7 Use a calculator to evaluate each to six decimal places:

(A) log 0.013 529

(B) ln 28.693 28

(C) ln(-0.438)

Solving logb x � y for x Find x to four decimal places, given the indicatedlogarithm:

(A)

(B) ln x = 2.386

log x = -2.315

(A) Change to equivalent exponential form.

Evaluate with a calculator.

(B) Change to equivalent exponential form.Evaluate with a calculator.

= 10.8699

x = e2.386

ln x = 2.386

= 0.0048

x = 10-2.315

log x = -2.315SOLUTION

E X A M P L E 8

MATCHED PROBLEM 8 Find x to four decimal places, given the indicated logarithm:

(A)

(B) log x = 2.0821

ln x = -5.062

Solving Exponential Equations Solve for x to four decimal places:

(A) (B) (C) 3x= 4ex

= 310x= 2

E X A M P L E 9



Exponential equations can also be solved graphically by graphing both sides ofan equation and finding the points of intersection. Figure 3 illustrates this approachfor the equations in Example 9.

(A) Take common logarithms of both sides.

Property 3Use a calculator.To four decimal places

(B) Take natural logarithms of both sides.Property 3Use a calculator.To four decimal places

(C) Take either natural or common logarithms of both sides. (Wechoose common logarithms.)Property 7Solve for x.

Use a calculator.

To four decimal places = 1.2619

x =

log 4

log 3

x log 3 = log 4

log 3x= log 4

3x= 4

= 1.0986

x = ln 3

ln ex= ln 3

ex= 3

= 0.3010

x = log 2

log 10x= log 2

10x= 2SOLUTION

�1

�2

5

2

(A) y2

� 2 y1

� 10x

�1

�2

5

2

(B) y2

� 3 y1

� ex

�1

�2

5

2

(C) y2

� 4 y1

� 3x

FIGURE 3 Graphical solution of exponential equations

MATCHED PROBLEM 9 Solve for x to four decimal places:

(A) (B) (C) 4x= 5ex

= 610x= 7

Discuss how you could find using either natural or common log-arithms on a calculator. [Hint: Start by rewriting the equation in exponentialform.]

y = log5 38.25Explore & Discuss 3

REMARK In the usual notation for natural logarithms, the simplifications of Example 4, parts (C)and (D) on page 114, become

With these formulas we can change an exponential function with base b, or a loga-rithmic function with base b, to expressions involving exponential or logarithmic func-tions, respectively, to the base e. Such change-of-base formulas are useful in calculus.

ex ln b= bx and

ln xln b

= logb x



APPLICATIONSA convenient and easily understood way of comparing different investments is touse their doubling times—the length of time it takes the value of an investment to dou-ble. Logarithm properties, as you will see in Example 10, provide us with just the righttool for solving some doubling-time problems.

Doubling Time for an Investment How long (to the next whole year) will ittake money to double if it is invested at 20% compounded annually?

We use the compound interest formula discussed in Section 2-4:

Compound interest

The problem is to find t, given and that is,

Solve for t by taking the natural or commonlogarithm of both sides (we choose thenatural logarithm).Property 7

Use a calculator.

[Note: (ln 2)/(ln 1.2) ln 2�ln 1.2]To the next whole year

When interest is paid at the end of 3 years, the money will not be doubled; when paidat the end of 4 years, the money will be slightly more than doubled.

L 4 years

Z L 3.8 years

t =

ln 2ln 1.2

t ln 1.2 = ln 2

ln 1.2t= ln 2

1.2t= 2

2 = 1.2t

2P = P(1 + 0.2)t

A = 2p;r = 0.20, m = 1,

A = Pa1 +

rmbmt

SOLUTION

E X A M P L E 1 0

Example 10 can also be solved graphically by graphing both sides of the equationand finding the intersection point (Fig. 4).2 = 1.2t,

0

0

4

8

FIGURE 4 y1 = 1.2x, y2 = 2

MATCHED PROBLEM 10 How long (to the next whole year) will it take money to triple if it is invested at 13%compounded annually?

It is interesting and instructive to graph the doubling times for various rates com-pounded annually. We proceed as follows:

t =

ln 2ln(1 + r)

t ln(1 + r) = ln 2

ln(1 + r)t= ln 2

(1 + r)t= 2

2 = (1 + r)t

2P = P(1 + r)t

A = P(1 + r)t



Figure 5 shows the graph of this equation (doubling time in years) for interest ratescompounded annually from 1 to 70% (expressed as decimals). Note the dramaticchange in doubling time as rates change from 1 to 20% (from 0.01 to 0.20).

FIGURE 5

t �ln 2

ln (1 � r)

10

20

30

40

50

60

70

0

Yea

rs

0.600.400.20

Rate compounded annually

r

t

Home Ownership Rates The U.S. Census Bureau published the data in Table 1on home ownership rates. Let x represent time in years with representing 1900.Use logarithmic regression to find the best model of the form ln x for thehome ownership rate y as a function of time x. Use the model to predict the homeownership rate in the United States in 2015 (to the nearest tenth of a percent).

y = a + bx = 0

Enter the data in a graphing calculator (Fig. 6A) and find the logarithmic regres-sion equation (Fig. 6B). The data set and the regression equation are graphed in Figure 6C. Using TRACE, we predict that the home ownership rate in 2015 would be69.4%.

SOLUTION

E X A M P L E 1 1

TABLE 1 Home Ownership Rates

Year Rate (%)

1950 55.0

1960 61.9

1970 62.9

1980 64.4

1990 64.2

2000 67.4

FIGURE 6

(A) (B)

40

40

80

120

(C)

Among increasing function, the logarithmic functions (with bases ) increasemuch more slowly for large values of x than either exponential or polynomial func-tions. When a visual inspection of the plot of a data set indicates a slowly increasingfunction, a logarithmic function often provides a good model.We use logarithmic re-gression on a graphing calculator to find the function of the form ln x thatbest fits the data.

y = a + b

b 7 1



CAUTION: Note that in Example 11 we let represent 1900. If we let rep-resent 1940, for example, we would obtain a different logarithmic regression equation,but the prediction for 2015 would be the same. We would not let represent1950 (the first year in Table 1) or any later year, because logarithmic functions areundefined at 0.

x = 0

x = 0x = 0

MATCHED PROBLEM 11 The home ownership rate in 2005 was 69.1%. Add this point to the data of Table 1,and, with representing 1900, use logarithmic regression to find the best modelof the form ln x for the home ownership rate y as a function of time x. Usethe model to predict the home ownership rate in the United States in 2015 (to thenearest tenth of a percent).

y = a + bx = 0

1.(A) (B) (C)

2.(A) (B) (C)

3.(A) (B) (C)

4.(A) (B) (C) (D)

5.

6.

7.(A) (B) 3.356 663 (C) Not defined

8.(A) 0.0063 (B) 120.8092

9.(A) 0.8451 (B) 1.7918 (C) 1.1610

10. 9 yr

11. 69.9%

-1.868 734

x = 5

x = 2

logb xbu23(logb R - logb S)logb R - logb S - logb T

b = 10x =13y =

32

log3(13) = -1log9 3 =

12log7 49 = 2

19 = 3-22 = 41/29 = 32Answers to Matched Problems

A For Problems 1–6, rewrite in equivalent exponential form.

1. 2.

3. 4.

5. 6.

For Problems 7–12, rewrite in equivalent logarithmic form.

7. 8.

9. 10.

11. 12.

For Problems 13–24, evaluate without a calculator.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24. log6 36log10 1,000

log3 35log2 2

-3

log10 10-5log10 103

log13 13log0.2 0.2

log10 10loge e

loge 1log10 1

M = bxA = bu

9 = 272/38 = 43/2

36 = 6249 = 72

log9 27 =32log4 8 =

32

loge 1 = 0log10 1 = 0

log2 32 = 5log3 27 = 3

For Problems 25–30, write in terms of simpler forms, as inExample 4.

25. 26.

27. 28.

29. 30.

B For Problems 31–42, find x, y, or b without a calculator.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42. logb 4 =23logb 1,000 =

32

log49(17) = ylog1/3 9 = y

log25 x =12log4 x =

12

logb e-2= -2logb 10-4

= -4

log3 27 = ylog7 49 = y

log2 x = 2log3 x = 2

log3 P

log3 R3p log3 q

logb w15logb L5

logb FGlogb P

Q

Exercise 2-5



In Problems 43–52, discuss the validity of each statement. If thestatement is always true, explain why. If not, give a counter-example.

43. Every polynomial function is one-to-one.

44. Every polynomial function of odd degree is one-to-one.

45. If g is the inverse of a function f, then g is one-to-one.

46. The graph of a one-to-one function intersects each verti-cal line exactly once.

47. The inverse of is

48. The inverse of is

49. If and then the graph of is in-creasing.

50. If then the graph of is increasing.

51. If f is one-to-one, then the domain of f is equal to therange of f.

52. If g is the inverse of a function f, then f is the inverse of g.

Find x in Problems 53–60.

53.

54.

55.

56.

57.

58.

59.

60.

Graph Problems 61 and 62 by converting to exponential formfirst.

61.

62.

63. Explain how the graph of the equation in Problem 61can be obtained from the graph of using asimple transformation (see Section 2-2).

64. Explain how the graph of the equation in Problem 62can be obtained from the graph of using asimple transformation (see Section 2-2).

65. What are the domain and range of the function definedby

66. What are the domain and range of the function definedby

For Problems 67 and 68, evaluate to five decimal places using acalculator.

y = log(x - 1) - 1?

y = 1 + ln(x + 1)?

y = log3 x

y = log2 x

y = log3(x + 2)

y = log2(x - 2)

log10(x + 6) - log10(x - 3) = 1

log10(x - 1) - log10(x + 1) = 1

logb(x + 2) + logb x = logb 24

logb x + logb(x - 4) = logb 21

logb x = 3 logb 2 +12 logb 25 - logb 20

logb x =32 logb 4 -

23 logb 8 + 2 logb 2

logb x =23 logb 27 + 2 logb 2 - logb 3

logb x =23 logb 8 +

12 logb 9 - logb 6

y = logb xb 7 1,

y = logb xb Z 1,b 7 0

g(x) = 1x.f(x) = x2

g(x) = x/2.f(x) = 2x

67. (A) log 3,527.2 (B) log 0.006 913 2

(C) ln 277.63 (D) ln 0.040 883

68. (A) log 72.604 (B) log 0.033 041

(C) ln 40,257 (D) ln 0.005 926 3

For Problems 69 and 70, find x to four decimal places.

69. (A) (B)

(C) (D)

70. (A) (B)

(C) (D)

For Problems 71–78, solve each equation to four decimal places.

71. 72.

73. 74.

75. 76.

77. 78.

Graph Problems 79–86 using a calculator and point-by-pointplotting. Indicate increasing and decreasing intervals.

79. 80.

81. 82.

83. 84.

85. 86.

C87. Explain why the logarithm of 1 for any permissible base

is 0.

88. Explain why 1 is not a suitable logarithmic base.

89. Write in an exponential formthat is free of logarithms.

90. Write in an exponential formthat is free of logarithms.

91. Let and Use a graph-ing calculator to draw graphs of all three functions in thesame viewing window for Discuss what itmeans for one function to be larger than another on aninterval, and then order the three functions from largestto smallest for

92. Let and Use agraphing calculator to draw graphs of all three functionsin the same viewing window for Discusswhat it means for one function to be smaller than anoth-er on an interval, and then order the three functionsfrom smallest to largest for 1 6 x … 16.

1 … x … 16.

r(x) = x.p(x) = log x, q(x) =31x,

1 6 x … 16.

1 … x … 16.

r(x) = x.p(x) = ln x, q(x) = 1x,

loge x - loge 25 = 0.2t

log10 y - log10 c = 0.8x

y = 4 ln(x - 3)y = 4 ln x - 3

y = 2 ln x + 2y = 2 ln(x + 2)

y = ln ƒ x ƒy = ƒ ln x ƒ

y = - ln xy = ln x

1.024t= 21.00512t

= 3

1.075x= 1.8371.03x

= 2.475

ex= 0.3059ex

= 4.304

10x= 15310x

= 12

ln x = -4.3281ln x = 3.1336

log x = -1.1577log x = 2.0832

ln x = -1.8879ln x = 2.7763

log x = -2.0497log x = 1.1285

Applications

93. Doubling time. In its first 10 years the Gabelli Growth Fundproduced an average annual return of 21.36%. Assume thatmoney invested in this fund continues to earn 21.36% com-pounded annually. How long will it take money invested in thisfund to double?

94. Doubling time. In its first 10 years the Janus Flexible IncomeFund produced an average annual return of 9.58%. Assumethat money invested in this fund continues to earn 9.58%compounded annually. How long will it take money investedin this fund to double?



95. Investing. How many years (to two decimal places) will it take$1,000 to grow to $1,800 if it is invested at 6% compoundedquarterly? Compounded daily?

96. Investing. How many years (to two decimal places) will it take$5,000 to grow to $7,500 if it is invested at 8% compoundedsemiannually? Compounded monthly?

97. Supply and demand. A cordless screwdriver is sold through anational chain of discount stores.A marketing company estab-lished price–demand and price–supply tables (Tables 1 and 2),where x is the number of screwdrivers people are willing to buyand the store is willing to sell each month at a price of p dollarsper screwdriver.

(A) Find a logarithmic regression model forthe data in Table 1. Estimate the demand (to the nearestunit) at a price level of $50.

(y = a + b ln x)

(B) Find a logarithmic regression model forthe data in Table 2. Estimate the supply (to the nearestunit) at a price level of $50.

(y = a + b ln x)

x p � D(x) ($)


1,000 91

2,000 73

3,000 64

4,000 56

5,000 53

x p � S(x) ($)

TABLE 2 Price–Supply

1,000 9

2,000 26

3,000 34

4,000 38

5,000 41

(C) Does a price level of $50 represent a stable condition, oris the price likely to increase or decrease? Explain.

98. Equilibrium point. Use the models constructed in Problem 97 tofind the equilibrium point. Write the equilibrium price to thenearest cent and the equilibrium quantity to the nearest unit.

99. Sound intensity: decibels. Because of the extraordinary range ofsensitivity of the human ear (a range of over 1,000 million mil-lions to 1), it is helpful to use a logarithmic scale, rather than anabsolute scale, to measure sound intensity over this range. Theunit of measure is called the decibel, after the inventor of the

telephone,Alexander Graham Bell. If we let N be the numberof decibels, I the power of the sound in question (in watts persquare centimeter), and the power of sound just below thethreshold of hearing (approximately watt per square cen-timeter), then

Show that this formula can be written in the form

100. Sound intensity: decibels. Use the formula in Problem 99 (with) to find the decibel ratings of the following

sounds:(A) Whisper:

(B) Normal conversation:

(C) Heavy traffic:

(D) Jet plane with afterburner:

101. Agriculture. Table 3 shows the yield (in bushels per acre) andthe total production (in millions of bushels) for corn in theUnited States for selected years since 1950. Let x representyears since 1900. Find a logarithmic regression model

for the yield. Estimate (to one decimal place)the yield in 2015.(y = a + b ln x)

10-1 W/cm2

10-8 W/cm2

3.16 * 10-10 W/cm2

10-13 W/cm2

I0 = 10-16 W/cm2

N = 10 log I

I0

I = I010N/10

10-16I0

Yield (bushels Total ProductionYear x per acre) (million bushels)

TABLE 3 United States Corn Production

1950 50 37.6 2,782

1960 60 55.6 3,479

1970 70 81.4 4,802

1980 80 97.7 6,867

1990 90 115.6 7,802

2000 100 139.6 10,192

102. Agriculture. Refer to Table 3. Find a logarithmic regressionmodel for the total production. Estimate (tothe nearest million) the production in 2015.

103. World population. If the world population is now 6.5 billionpeople and if it continues to grow at an annual rate of 1.14%compounded continuously, how long (to the nearest year) willit take before there is only 1 square yard of land per person?(The Earth contains approximately square yards ofland.)

104. Archaeology: carbon-14 dating. The radioactive carbon-14in an organism at the time of its death decays according

to the equation

where t is time in years and is the amount of present attime (See Example 3 in Section 2-4.) Estimate the ageof a skull uncovered in an archaeological site if 10% of theoriginal amount of is still present. [Hint: Find t such thatA = 0.1A0.

14C

t = 0.

14CA0

A = A0e-0.000124t

(14C)

1.68 * 1014

(y = a + b ln x)


Chapter 2 Review 119

CHAPTER 2 REVIEW

2-1 Functions

• Point-by-point plotting may be used to sketch the graph of an equation in two variables: Plot enoughpoints from its solution set in a rectangular cordinate system so that the total graph is apparent andthen connect these points with a smooth curve.

• A function is a correspondence between two sets of elements such that to each element in the first setthere corresponds one and only one element in the second set. The first set is called the domain andthe set of corresponding elements in the second set is called the range.

• If x is a placeholder for the elements in the domain of a function, then x is called the independentvariable or the input. If y is a placeholder for the elements in the range, then y is called the dependentvariable or the output.

• If in an equation in two variables we get exactly one ouput for each input, then the equation specifiesa function. The graph of such a function is just the graph of the specifying equation. If we get morethan one output for a given input, then the equation does not specify a function.

• The vertical-line test can be used to determine whether or not an equation in two variables specifies afunction (Theorem 1, p. 51).

• The functions specified by equations of the form where are called linear func-tions. Functions specified by equations of the form are called constant functions.

• If a function is specified by an equation and the domain is not indicated, we agree to assume that thedomain is the set of all inputs that produce outputs that are real numbers.

• The symbol f(x) represents the element in the range of f that corresponds to the element x of thedomain.

• Break-even and profit–loss analysis use a cost function C and a revenue function R to determinewhen a company will have a loss will break even or will have a profit Typical cost, revenue, profit, and price–demand functions are given on page 55.

2-2 Elementary Functions: Graphs and Transformations

• The graphs of six basic elementary functions (the identity function, the square and cube functions, thesquare root and cube root functions, and the absolute value function) are shown on page 65.

• Performing an operation on a function produces a transformation of the graph of the function. Thebasic graph transformations, vertical and horizontal translations (shifts), reflection in the x axis, andvertical stretches and shrinks, are summarized on page 69.

• A piecewise-defined function is a function whose definition involves more than one formula.

2-3 Quadratic Functions

• If a, b, and c are real numbers with then the function

Standard form

is a quadratic function in standard form and its graph is a parabola.

• The quadratic formula

can be used to find the x intercepts of a quadratic function.

• Completing the square in the standard form of a quadratic function produces the vertex form

Vertex form

• From the vertex form of a quadratic function, we can read off the vertex, axis of symmetry, maximumor minimum, and range, and can easily sketch the graph (page 81).

• If a revenue function and a cost function intersect at a point then both this pointand its x coordinate are referred to as break-even points.x0

(x0, y0),C(x)R(x)

f(x) = a(x - h)2+ k

b2- 4ac Ú 0x =

- b ; 2b2- 4ac

2a

f(x) = ax2+ bx + c

a Z 0,

(R 7 C).(R = C),(R 6 C),

y = bm Z 0,y = mx + b,

Important, Terms, Symbols, and Concepts

Ex. 1, p. 47

Ex. 2, p. 50

Ex. 3, p. 52Ex. 5, p. 54

Ex. 4, p. 53Ex. 6, p. 55

Ex. 7, p.56

Ex. 1, p. 64

Ex. 2, p. 66Ex. 3, p. 67Ex. 4, p. 68Ex. 5, p. 69Ex. 6, p. 70

Ex. 1, p. 77

Ex. 2, p. 81Ex. 3, p. 83

Ex. 4, p. 84

Examples


• Quadratic regression on a graphing calculator produces the function of the form that best fits a data set.

• A quadratic function is a special case of a polynomial function, that is, a function that can be writtenin the form

for n a nonnegative integer called the degree of the polynomial. The coefficients are realnumbers with . The domain of a polynomial function is the set of all real numbers. Graphs ofrepresentative polynomial functions are shown on page 87 and inside the front cover.

• The graph of a polynomial function of degree n can intersect the x axis at most n times.

• The graph of a polynomial function has no sharp corners and is continuous, that is, it has no holes orbreaks.

• A rational function is any function that can be written in the form

where and are polynomials. The domain is the set of all real numbers such that Graphs of representative rational functions are shown on page 88 and inside the front cover.

• Unlike polynomial functions, a rational function can have vertical asymptotes [but not more than thedegree of the denominator ] and at most one horizontal asymptote.

2-4 Exponential Functions

• An exponential function is a function of the form

where is a positive constant called the base. The domain of f is the set of all real numbers andthe range is the set of positive real numbers.

• The graph of an exponential function is continuous, passes through (0, 1), and has the x axis as ahorizontal asymptote. If then increases as x increases; if then decreases as xincreases (Theorem 1, p. 95).

• Exponential functions obey the familiar laws of exponents and satisfy additional properties(Theorem 2, p. 96).

• The base that is used most frequently in mathematics is the irrational number .

• Exponential functions can be used to model population growth and radioactive decay.

• Exponential regression on a graphing calculator produces the function of the form that bestfits a data set.

• Exponential functions are used in computations of compound interest:

Compound interest formula

(see summary on page 102).

2-5 Logarithmic Functions

• A function is said to be one-to-one if each range value corresponds to exactly one domain value.

• The inverse of a one-to-one function f is the function formed by interchanging the independent anddependent variables of f. That is, (a, b) is a point on the graph of f if and only if (b, a) is a point on thegraph of the inverse of f. A function that is not one-to-one does not have an inverse.

• The inverse of the exponential function with base b is called the logarithmic function with base b,denoted The domain of is the set of all positive real numbers (which is the range of

), and the range of is the set of all real numbers (which is the domain of ).

• Because is the inverse of the function

Logarithmic form Exponential formis equivalent to x = byy = logb x

bx,logb x

bxlogb xbxlogb xy = logb x.

A = Pa1 +

r

mbmt

y = abx

e L 2.7183

bx0 6 b 6 1,bxb 7 1,

b Z 1

f(x) = bx

d(x)

d(x) Z 0.d(x)n(x)

d(x) Z 0f(x) =

n(x)

d(x)

an Z 0a0, a1, . . . , an

f(x) = anxn+ an - 1x

n - 1+ ... + a1x + a0

y = ax2+ bx + c Ex. 5, p. 86

Ex. 1, p. 95

Ex. 2, p. 98

Ex. 3, p. 98

Ex. 4, p. 100

Ex. 5, p. 101

Ex. 1, p. 108Ex. 2, p. 108Ex. 3, p. 109



• Properties of logarithmic functions can be obtained from corresponding properties of expontial func-tions (Theorem 1, p. 109).

• Logarithms to the base 10 are called common logarithms, often denoted simply by log x. Logarithms tothe base e are called natural logarithms, often denoted by ln x.

• Logarithms can be used to find an investment’s doubling time—the length of time it takes for the valueof an investment to double.

• Logarithmic regression on a graphing calculator produces the function of the form that best fits a data set.

y = a + b ln x

Ex. 4, p.110Ex. 5, p. 110Ex. 6, p. 111Ex. 7, p. 111Ex. 8, p. 112Ex. 9, p. 112

Ex. 10, p. 114

Ex. 11, p. 115

REVIEW EXERCISE

Work through all the problems in this chapter review andcheck your answers in the back of the book. Answers to all re-view problems are there along with section numbers in italicsto indicate where each type of problem is discussed. Whereweaknesses show up, review appropriate sections in the text.

A In Problems 1–3, use point-by-point plotting to sketch thegraph of each equation.

1.2.3.4. Indicate whether each graph specifies a function:

(A) (B)

(C) (D)

5. For and , find:(A) (B)

(C) (D)f(3)

g(2)

g(2)

f(3)

f(0) # g(4)f(-2) + g(-1)g(x) = x2

- 2xf(x) = 2x - 1

y2= 4x2

x2= y2

y = 5 - x2

6. Write in logarithmic form using base e:7. Write in logarithmic form using base 10:8. Write in exponential form using base e:9. Write in exponential form using base 10:

Simplify Problems 10 and 11.

10. 11.

Solve Problems 12–14 for x exactly without using a calculator.

12. 13.14.

Solve Problems 15–18 for x to three decimal places.

15. 16.17. 18.

19. Use the graph of function f in the figure to determine (tothe nearest integer) x or y as indicated.(A) (B)(C) (D)(E) (F) -1 = f(x)y = f(-6)

3 = f(x)y = f(3)4 = f(x)y = f(0)

ln x = -1.147log x = 3.105ex

= 503,00010x= 143.7

log2 16 = x

logx 36 = 2log3 x = 2

a eu

e-u bu5x + 4

54 - x

log u = v.ln M = N.x = 10y.

u = ev.

y

x

�5

5�5

5

y

x

�5

5�5

5

y

x

�5

5�5

5

y

x

�5

5�5

5

x

f (x)

�5

7�7

5

Figure for 19




20. Sketch a graph of each of the functions in parts (A)–(D)using the graph of function f in the figure below.(A) (B)(C) (D) y = -f(x + 3) - 3y = f(x - 2)

y = f(x) + 4y = -f(x)

formation for you to sketch the graph of ? Addmore points to the table until you are satisfied that yoursketch is a good representation of the graph of on the interval [ ].

x 0 2

f(x) 0 4

In Problems 26–29, each equation specifies a function. Deter-mine whether the function is linear, quadratic, constant, ornone of these.

26. 27.

28. 29.

Solve Problems 30–37 for x exactly without using a calculator.

30. 31.32. 33.34. 35.36. 37.

Solve Problems 38–47 for x to four decimal places.

38. 39.40. 41. ln 42. 43.44. 45.46. 47.48. Find the domain of each function:

(A) (B)

49. The function g is defined by Translateinto a verbal definition.

50. Find the vertex form for and thenfind the intercepts, the vertex, the maximum or minimum,and the range.

51. Let and Find all pointsof intersection for the graphs of f and g. Round answers totwo decimal places.

In Problems 52 and 53, use point-by-point plotting to sketchthe graph of each function.

52.

53.

If , find and simplify each of the following in Problems 54–57.

54.55.56.57.58. Let Find

(A) f(2) (B)

(C) (D)f(2 + h) - f(2)

hf(2 + h) - f(2)

f(2 + h)f(x) = 3 - 2x.

f(4 - x)f(2x - 1)f(f(-1))f(f(0))

f(x) = 5x + 1

f(x) =

-66

2 + x2

f(x) =

50

x2+ 1

g(x) = ln (x + 2).f(x) = ex- 1

f(x) = 4x2+ 4x - 3

g(x) = 2x - 31x.

g(x) =

3x

25 - xf(x) =

2x - 5

x2- x - 6

x = log0.2 5.321x = log2 752x - 3

= 7.088,000 = 4,000(1.08x)0.01 = e-0.05x35 = 7(3x)

x = 0.3618log x = -2.0144x = 230(10-0.161)x = 3(e1.492

log9 x =32logx 8 = -3

log1/3 9 = x2x2ex= 3xex

e2x= ex2

- 39x - 1= 31 + x

2 ln(x - 1) = ln(x2- 5)log(x + 5) = log(2x - 3)

y = 8x + 2(10 - 4x)y =

7 - 4x

2x

y =

1 + 5x

6y = 4 - x + 3x2

-4

-2

-5, 5y = f(x)

y = f(x)

x

�5

5�5

5

f (x)

Figure for 20

21. Complete the square and find the standard form for thequadratic function

Then write a brief verbal description of the relationshipbetween the graph of f and the graph of .

22. Match each equation with a graph of one of the functionsf, g, m, or n in the figure.

y = x2

f(x) = -x2+ 4x

x

y

�5

5�5

5

f g

m n

Figure for 22

(A) (B)(C) (D)

23. Referring to the graph of function f in the figure for Prob-lem 11 and using known properties of quadratic functions,find each of the following to the nearest integer:(A) Intercepts (B) Vertex(C) Maximum or minimum (D) Range(E) Increasing interval (F) Decreasing interval

24. Consider the set S of people living in the town of Newville.Which of the following correspondences specify a func-tion? Explain.(A) Each person in Newville (input) is paired with his or

her mother (output).

(B) Each person in Newville (input) is paired with his orher children (output).

25. The three points in the table are on the graph of the indi-cated function f. Do these three points provide sufficient in-

y = (x + 2)2- 4y = -(x - 2)2

+ 4y = -(x + 2)2

+ 4y = (x - 2)2- 4

f(x) = 6x - x3



59. Let Find(A) f(a) (B)

(C) (D)

60. Explain how the graph of is related tothe graph of

61. Explain how the graph of is related tothe graph of .

62. The following graph is the result of applying a sequence oftransformations to the graph of . Describe the trans-formations verbally and write an equation for the graph.

y = x2

y = x3g(x) = 0.3x3

+ 3y = ƒ x ƒ .

m(x) = - ƒ x - 4 ƒ

f(a + h) - f(a)

hf(a + h) - f(a)

f(a + h)f(x) = x2

- 3x + 1. 77. Write an equation for the graph shown in the formwhere a is either or and h and

k are integers.+1-1y = a(x - h)2

+ k,

x

y

�5

5�5

5

Figure for 62

63. The graph of a function f is formed by vertically stretchingthe graph of by a factor of 2, and shifting it to theleft 3 units and down 1 unit. Find an equation for functionf and graph it for and

In Problems 64–71, discuss the validity of each statement. If thestatement is always true, explain why. If not, give a counter-example.

64. Every polynomial function is a rational function.65. Every rational function is a polynomial function.66. The graph of every rational function has at least one verti-

cal asymptote.67. The graph of every exponential function has a horizontal

asymptote.68. The graph of every logarithmic function has a vertical as-

ymptote.69. There exists a logarithmic function that has both a vertical

and horizontal asymptote.70. There exists a rational function that has both a vertical and

horizontal asymptote.71. There exists an exponential function that has both a verti-

cal and horizontal asymptote.

Graph Problems 72–74 over the indicated interval. Indicateincreasing and decreasing intervals.

72. 73.74.75. Sketch the graph of f for

76. Sketch the graph of g for

g(x) = c 0.5x + 5 if 0 … x … 101.2x - 2 if 10 6 x … 30

2x - 26 if x 7 30

x Ú 0.

f(x) = e9 + 0.3x if 0 … x … 205 + 0.2x if x 7 20

x Ú 0.y = ln(x + 1); (-1, 10]

f(t) = 10e-0.08t; t Ú 0y = 2x - 1; [-2, 4]

-5 … y … 5.-5 … x … 5

y = 1 x

x

y

�5

�5

5

Figure for 77

78. Given , find the following al-gebraically (to three decimal places) without referring to agraph:(A) Intercepts (B) Vertex(C) Maximum or minimum (D) Range

79. Graph in a graphing calcu-lator and find the following (to three decimal places)using TRACE and appropriate built-in commands:(A) Intercepts (B) Vertex(C) Maximum or minimum (D) Range

C 80. Noting that andexplain why the calculator results

shown here are obvious. Discuss similar connections be-tween the natural logarithmic function and the exponentialfunction with base e.

12 = 1.414 213 562 . . . p = 3.141 592 654 . . .

f(x) = -0.4x2+ 3.2x + 1.2

f(x) = -0.4x2+ 3.2x + 1.2

Solve Problems 81–84 exactly without using a calculator.

81.82.83.84.85. Write in an exponential form free of log-

arithms.Then solve for y in terms of the remaining variables.86. Explain why 1 cannot be used as a logarithmic base.

87. The following graph is the result of applying a sequence oftransformations to the graph of . Describe the trans-formations verbally, and write an equation for the graph.

y =31x

ln y = -5t + ln c

log 3x2= 2 + log 9x

ln(x + 3) - ln x = 2 ln 2

ln(2x - 2) - ln(x - 1) = ln x

log x - log 3 = log 4 - log(x + 4)

x

y

�5

5�5

5

Figure for 87



88. Given , find the following alge-braically (to three decimal places) without the use of a graph:(A) Intercepts

(B) Vertex


(D) Range

(E) Increasing and decreasing intervals

G(x) = 0.3x2+ 1.2x - 6.9

A mathematical model for this data is given by

where corresponds to 1980.

(A) Complete the following table.

x = 0

f(x) = 0.181x2- 4.88x + 271

(B) Graph S(x).

Year Number of Eggs

1980 271

1985 255

1990 233

1995 236

2000 256

2005 257

89. Graph in a standard viewingwindow. Then find each of the following (to three decimalplaces) using appropriate commands.(A) Intercepts

(B) Vertex


(D) Range

(E) Increasing and decreasing intervals

G(x) = 0.3x2+ 1.2x - 6.9

x Consumption f(x)

0 271

5 255

10 233

15 236

20 256

25 257

In all problems involving days, a 365-day year is assumed.

90. Egg consumption. The table shows the annual per capita con-sumption of eggs in the United States.

Energy Charge (June–September)

$3.00 for the first 20 kWh or less

5.70¢ per kWh for the next 180 kWh

3.46¢ per kWh for the next 800 kWh

2.17¢ per kWh for all over 1,000 kWh

(B) Graph and the data in the same coordinatesystem.

(C) Use the modeling function f to estimate the per capitaegg consumption in 2012.

(D) Based on the information in the table, write a brief de-scription of egg consumption from 1980 to 2005.

91. Egg consumption. Using quadratic regression on a graphingcalculator, show that the quadratic function that best fits thedata on egg consumption in Problem 90 is

(coefficients rounded to three significant digits).92. Electricity rates. The table shows the electricity rates charged

by Easton Utilities in the summer months.(A) Write a piecewise definition of the monthly charge S(x)

(in dollars) for a customer who uses x kWh in a summermonth.

f(x) = 0.181x2- 4.88x + 271

y = f(x)

Applications

93. Money growth. Provident Bank of Cincinnati, Ohio recentlyoffered a certificate of deposit that paid 5.35% compoundedquarterly. If a $5,000 CD earns this rate for 5 years, how muchwill it be worth?

94. Money growth. Capital One Bank of Glen Allen, Virginiarecently offered a certificate of deposit that paid 4.82% com-pounded daily. If a $5,000 CD earns this rate for 5 years, howmuch will it be worth?

95. Money growth. How long will it take for money invested at6.59% compounded monthly to triple?

96. Money growth. How long will it take for money invested at7.39% compounded daily to double?

97. Break-even analysis. The research department in a companythat manufactures AM/FM clock radios established the fol-lowing price–demand, cost, and revenue functions:

Price–demand function

Cost function

Revenue function

where x is in thousands of units, and C(x) and R(x) are inthousands of dollars. All three functions have domain

.(A) Graph the cost function and the revenue function simu-

taneously in the same coordinate system.

(B) Determine algebraically when . Then, with the aidof part (A), determine when and to thenearest unit.

(C) Determine algebraically the maximum revenue (to thenearest thousand dollars) and the output (to the nearestunit) that produces the maximum revenue.What is thewholesale price of the radio (to the nearest dollar) at thisoutput?

98. Profit–loss analysis. Use the cost and revenue functions fromProblem 97.(A) Write a profit function and graph it in a graphing

calculator.

R 7 CR 6 CR = C

1 … x … 40

= x(50 - 1.25x)

R(x) = xp(x)

C(x) = 160 + 10x

p(x) = 50 - 1.25x



(A) Find an exponential regression model forthis data. Estimate (to the nearest million) the numberof subscribers in 2015.

(B) Some analysts estimate the number of wirelesssubscribers in 2018 to be 300 million. How does thiscompare with the prediction of the model of part(A)? Explain why the model will not give reliablepredictions far into the future.

102. Medicine. One leukemic cell injected into a healthymouse will divide into 2 cells in about day. At theend of the day these 2 cells will divide into 4. Thisdoubling continues until 1 billion cells are formed; thenthe animal dies with leukemic cells in every part of the body.(A) Write an equation that will give the number N of

leukemic cells at the end of t days.

(B) When, to the nearest day, will the mouse die?

12

(y = abx)

x

y y

x p � D(x) ($)


985 330

2,145 225

2,950 170

4,225 105

5,100 50

x p � S(x) ($)

TABLE 2 Price–Supply

985 30

2,145 75

2,950 110

4,225 155

5,100 190

(A) Find a quadratic regression model for the data inTable 1. Estimate the demand at a price level of $180.

(B) Find a linear regression model for the data in Table 2. Estimate the supply at a price level of $180.

(C) Does a price level of $180 represent a stable condition,or is the price likely to increase or decrease? Explain.

(D) Use the models in parts (A) and (B) to find theequilibrium point. Write the equilibrium priceto the nearest cent and the equilibrium quantityto the nearest unit.

101. Telecommunications. According to the TelecommunicationsIndustry Association, wireless telephone subscriptions grewfrom about 4 million in 1990 to over 180 million in 2005(Table 3). Let x represent years since 1990.

Year Million Subscribers

1990 4

1993 13

1996 38

1999 76

2002 143

2005 180

TABLE 3 Wireless Telephone Subscribers

(B) Determine graphically when , , and to the nearest unit.

(C) Determine graphically the maximum profit (to the near-est thousand dollars) and the output (to the nearest unit)that produces the maximum profit. What is the wholesaleprice of the radio (to the nearest dollar) at this output?[Compare with Problem 97C.]

99. Construction. A construction company has 840 feet of chain-link fence that is used to enclose storage areas for equipmentand materials at construction sties. The supervisor wants to setup two identical rectangular storage areas sharing a commonfence (see the figure).

P 7 0P 6 0P = 0

Assuming that all fencing is used,(A) Express the total area A(x) enclosed by both pens as a

function of x.(B) From physical considerations, what is the domain of the

function A?(C) Graph function A in a rectangular coordinate system.(D) Use the graph to discuss the number and approximate

locations of values of x that would produce storageareas with a combined area of 25,000 square feet.

(E) Approximate graphically (to the nearest foot) the val-ues of x that would produce storage areas with a com-bined area of 25,000 square feet.

(F) Determine algebraically the dimensions of the storageareas that have the maximum total combined area.What is the maximum area?

100. Equilibrium point. A company is planning to introducea 10-piece set of nonstick cookware. A marketing companyestablished price–demand and price–supply tables forselected prices (Tables 1 and 2), where x is the numberof cookware sets people are willing to buy and the companyis willing to sell each month at a price of p dollars per set.

103. Marine biology. The intensity of light entering water isreduced according to the exponential equation

where I is the intensity d feet below the surface, I0 is theintensity at the surface, and k is the coefficient of extinction.Measurements in the Sargasso Sea in the West Indies haveindicated that half of the surface light reaches a depth of

I = I0e-kd



73.6 feet. Find k (to five decimal places), and find the depth(to the nearest foot) at which 1% of the surface lightremains.

104. Agriculture. The number of dairy cows on farms in theUnited States is shown in Table 4 for selected years since1950. Let 1940 be year 0.

Year Billion $

TABLE 5 Medicare Expenditures

1980 37

1985 72

1990 111

1995 181

2000 197

2005 330

Year Dairy Cows(thousands)

TABLE 4 Dairy Cows on Farms in the United States

1950 23,853

1960 19,527

1970 12,091

1980 10,758

1990 10,015

2000 9,190

(A) Find a logarithmic regression model for the data. Estimate (to the nearest thousand) thenumber of dairy cows in 2015.

(B) Explain why it is not a good idea to let 1950 be year 0.

(y = a + b ln x)

105. Population growth. Some countries have a relative growthrate of 3% (or more) per year.At this rate, how many years (tothe nearest tenth of a year) will it take a population to double?

106. Medicare. The annual expenditures for Medicare (in bil-lions of dollars) by the U.S. government for selected yearssince 1980 are shown in Table 5. Let x represent years since1980.

(A) Find an exponential regression model for thedata. Estimate (to the nearest billion) the annual ex-penditures in 2015.

(B) When will the annual expenditures reach one trilliondollars?

(y = abx)


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