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Functions and Their Graphs

Chapter 2

Functions

Section 2.1

Relations

Relation: A correspondence between two sets.

x corresponds to y or y depends on x

if a relation exists between x and y

Denote by x → y in this case.

Relations

Example.

Melissa

John

Jennifer

Patrick

$45,000

$40,000

$50,000

Person Salary

Relations

Example.

0

1

4

0

1

—1

2

—2

Number Number

Functions

Function: special kind of relation

Each input corresponds to precisely one output

If X and Y are nonempty sets, a function from X into Y is a relation that associates with each element of Xexactly one element of Y

Functions

Example.Problem: Does this relation represent a function?

Answer:

Melissa

John

Jennifer

Patrick

$45,000

$40,000

$50,000

Person Salary

Functions

Example.Problem: Does this relation represent a function?

Answer:

0

1

4

0

1

—1

2

—2

Number Number

Domain and Range

Function from X to Y

Domain of the function: the set X.

If x in X:

The image of x or the value of the function at x: The element y corresponding to x

Range of the function: the set of all values of the function

Domain and Range

Example.Problem: What is the range of this function?

Answer:

0

1

4

9

—3

—2

—1

0

1

2

3

X Y

Domain and Range

Example. Determine whether the relation represents a function. If it is a function, state the domain and range.

Problem:

Relation: {(2,5), (6,3), (8,2), (4,3)}

Answer:

Domain and Range

Example. Determine whether the relation represents a function. If it is a function, state the domain and range.

Problem:

Relation: {(1,7), (0, —3), (2,4), (1,8)}

Answer:

Equations as Functions

To determine whether an equation is a function

Solve the equation for y.

If any value of x in the domain corresponds to more than one y, the equation doesn’t define a function

Otherwise, it does define a function.


Example.

Problem: Determine if the equation

x + y2 = 9

defines y as a function of x.

Answer:

Function as a Machine

Accepts numbers from domain as input.

Exactly one output for each input.

Finding Values of a Function

Example. Evaluate each of the following for the function

f(x) = —3x2 + 2x

(a) Problem: f(3)

Answer:

(b) Problem: f(x) + f(3)

Answer:

(c) Problem: f(—x)

Answer:

(d) Problem: —f(x)

Answer:

(e) Problem: f(x+3)

Answer:


Example. Evaluate the difference

quotient

of the function

Problem: f(x) = — 3x2 + 2x.

Answer:

Implicit Form of a Function

A function given in terms of xand y is given implicitly.

If we can solve an equation for yin terms of x, the function is given explicitly


Example. Find the explicit form of the implicit function.

(a) Problem: 3x + y = 5

Answer:

(b) Problem: xy + x = 1

Answer:

Important Facts

For each x in the domain of f, there is exactly one image f(x) in the range

An element in the range can result from more than one x in the domain

We usually call x the independentvariable

y is the dependent variable

Finding the Domain

If the domain isn’t specified, it will always be the largest set of real numbers for which f(x) is a real number

We can’t take square roots of negative numbers (yet) or divide by zero

Finding the Domain

Example. Find the domain of each of the following functions.

(a) Problem: f(x) = x2 — 9

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:

Finding the Domain

Example. A rectangular garden has a perimeter of 100 feet.

(a) Problem: Express the area A of the

garden as a function of the width w.

Answer:

(b) Problem: Find the domain of A(w)

Answer:

Operations on Functions

Arithmetic on functions f and g

Sum of functions:

(f + g)(x) = f(x) + g(x)

Difference of functions:

(f — g)(x) = f(x) — g(x)

Domains: Set of all real numbers in the domains of both f and g.

For both sum and difference


Arithmetic on functions f and g

Product of functions f and g is

(f · g)(x) = f(x) · g(x)

The quotient of functions f and g is

Domain of product: Set of all real numbers in the domains of both f and g

Domain of quotient: Set of all real numbers in

the domains of both f and g with g(x) ≠ 0

)(

)()(

xg

xfx

g

f=⎟⎟

⎠

⎞⎜⎜⎝

⎛


Example. Given f(x) = 2x2 + 3 and g(x) = 4x3 + 1.

(a) Problem: Find f+g and its domain

Answer:

(b) Problem: Find f — g and its domain

Answer:


Example. Given f(x) = 2x2 + 3 and g(x) = 4x3 + 1.

(c) Problem: Find f·g and its domain

Answer:

(d) Problem: Find f/g and its domain

Answer:

Key Points

Relations

Functions

Domain and Range


Function as a Machine



Important Facts

Finding the Domain

Key Points (cont.)


The Graph of a Function

Section 2.2

Vertical-line Test

Theorem. [Vertical-Line Test]

A set of points in the xy-plane is the

graph of a function if and only if

every vertical line intersects the

graphs in at most one point.

Vertical-line Test

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Example.

Problem: Is the graph that of a function?

Answer:

Finding Information From Graphs

Example. Answer the questions about the graph.

(a) Problem: Find f(0)

Answer:

(b) Problem: Find f(2)

Answer:

(c) Problem: Find the domain

Answer:

(d) Problem: Find the range

Answer:

-4 -2 2 4

-4

-2

2

4

H2,4ÄÄÄÄÄ5LH-2, 4

ÄÄÄÄÄ5L

H1,2LH-1,2L

H0,4L



(e) Problem: Find the

x-intercepts:

Answer:

(f) Problem: Find the

y-intercepts:

Answer:

-4 -2 2 4

-4

-2

2

4


ÄÄÄÄÄ5L

H1,2LH-1,2L

H0,4L



(g) Problem: How often does

the line y = 3 intersect the

graph?

Answer:

(h) Problem: For what values

of x does f(x) = 2?

Answer:

(i) Problem: For what values

of x is f(x) > 0?

Answer:

-4 -2 2 4

-4

-2

2

4


ÄÄÄÄÄ5L

H1,2LH-1,2L

H0,4L

Finding Information From Formulas

Example. Answer the following questions for the function

f(x) = 2x2 — 5(a) Problem: Is the point (2,3) on the graph of

y = f(x)?

Answer:

(b) Problem: If x = —1, what is f(x)? What is the corresponding point on the graph?

Answer:

(c) Problem: If f(x) = 1, what is x? What is (are) the corresponding point(s) on the graph?

Answer:

Key Points

Vertical-line Test


Finding Information From Formulas

Properties of Functions

Section 2.3

Even and Odd Functions

Even function:

For every number x in its domain, the

number —x is also in the domain

f(—x) = f(x)

Odd function:

For every number x in its domain, the

number —x is also in the domain

f(—x) = —f(x)

Description of Even and Odd Functions

Even functions:

If (x, y) is on the graph, so is (—x, y)

Odd functions:

If (x, y) is on the graph, so is (—x, —y)


Theorem. A function is even if and only if its graph is symmetric with respect to the y-axis.A function is odd if and only if its graph is symmetric with respect to the origin.


Example.

Problem: Does the graph represent a function which is even, odd, or neither?

Answer:

-4 -2 2 4

-4

-2

2

4

Identifying Even and Odd Functions from the Equation

Example. Determine whether the following functions are even, odd or neither.

(a) Problem:

Answer:

(b) Problem: g(x) = 3x2 — 4

Answer:

(c) Problem:

Answer:

Increasing, Decreasing and Constant Functions

Increasing function (on an open interval I).

For any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2)

Decreasing function (on an open interval I)

For any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2)

Constant function (on an open interval I) For all choices of x in I, the values f(x) are equal.


Example. Answer the questions about the function shown.(a) Problem: Where is the

function increasing?

Answer:

(b) Problem: Where is the function decreasing?

Answer:

(c) Problem: Where is the function constant

Answer:

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6


WARNING!

Describe the behavior of a graph in terms of its x-values.

Answers for these questions should be open intervals. -6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Local Extrema

Local maximum at c:

Open interval I containing x so that, for all x ≤ c in I, f(x) ≤ f(c).

f(c) is a local maximum of f.

Local minimum at c:

Open interval I containing x so that, for all x ≤ c in I, f(x) ≥ f(c).

f(c) is a local minimum of f.

Local extrema.

Collection of local maxima and minima

Local Extrema

For local maxima:

Graph is increasing to the left of c

Graph is decreasing to the right of c.

For local minima:

Graph is decreasing to the left of c

Graph is increasing to the right of c.

Local Extrema

Example. Answer the questions about the given graph of f.

(a) Problem: At which

number(s) does f have a

local maximum?

Answer:

(b) Problem: At which

number(s) does f have a

local minimum?

Answer:

-7.5 -5 -2.5 2.5 5 7.5

-6

-4

-2

2

4

6

Average Rate of Change

Slope of a line can be interpreted as the average rate of change

Average rate of change: If c is in the domain of y = f(x)

Also called the difference quotient of fat c


Example. Find the average rates of change of

(a) Problem: From 0 to 1.

Answer:

(b) Problem: From 0 to 3.

Answer:

(c) Problem: From 1 to 3:

Answer:

Secant Lines

Geometric interpretation to the average rate of change

Label two points (c, f(c)) and (x, f(x))

Draw a line containing the points.

This is the secant line.

Theorem. [Slope of the Secant Line]The average rate of change of a function equals the slope of the secant line containing two points on its graph

Secant Lines

-7.5 -5 -2.5 2.5 5 7.5

-5

-2.5

2.5

5

7.5

10

12.5

15

Secant Lines

Example.

Problem: Find an

equation of the

secant line to

containing (0, f(0))

and (5, f(5))

Answer:

Key Points

Even and Odd Functions


Identifying Even and Odd Functions from the Equation


Local Extrema


Key Points (cont.)

Secant Lines

Linear Functions and Models

Section 2.4

Linear Functions

Linear function:

Function of the form f(x) = mx + b

Graph: Line with slope m and y-intercept b.

Theorem. [Average Rate of Change of

Linear Function]

Linear functions have a constant average

rate of change. The constant average rate

of change of f(x) = mx + b is

-10 -5 5 10

-10

-7.5

-5

-2.5

2.5

5

7.5

10

Linear Functions

Example.

Problem: Graph the linear function

f(x) = 2x — 5

Answer:

Application: Straight-Line Depreciation

Example. Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The company chooses to depreciate the machine using the straight-line method over 10 years.For straight-line depreciation, the value of the asset declines by a fixed amount every year.

2 4 6 8 10 12 14

-40000

-20000

20000

40000

60000

80000100000

120000

140000

Example. (cont.)

(a) Problem: Write a linear function that expresses the book value of the machine as a function of its age, x

Answer:

(b) Problem: Graph the linear function

Answer:


Example. (cont.)

(c) Problem: What is the book value of the machine after 4 years?

Answer:

(d) Problem: When will the machine be worth $20,000?

Answer:


Scatter Diagrams

Example. The amount of money that a lending institution will allow you to borrow mainly depends on the interest rate and your annual income.

The following data represent the annual income, I, required by a bank in order to lend L dollars at an interest rate of 7.5% for 30 years.

Scatter Diagrams

Example. (cont.) Annual Income, I ($)

Loan Amount, L ($)

15,000 44,600

20,000 59,500

25,000 74,500

30,000 89,400

35,000 104,300

40,000 119,200

45,000 134,100

50,000 149,000

55,000 163,900

60,000 178,800

65,000 193,700

70,000 208,600

Scatter Diagrams

Example. (cont.)

Problem: Use a graphing utility to draw a scatter diagram of the data.

Answer:

Linear and Nonlinear Relationships

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

6 .0

7 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

6 .0

7 .0

8 .0

9 .0

1 0 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 0 .0

-8 .0

-6 .0

-4 .0

-2 .0

0 .0

2 .0

4 .0

6 .0

8 .0

1 0 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 2 .0

-1 0 .0

-8 .0

-6 .0

-4 .0

-2 .0

0 .0

2 .0

4 .0

6 .0

8 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 5 .0

-1 0 .0

-5 .0

0 .0

5 .0

1 0 .0

1 5 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 5 .0

-1 0 .0

-5 .0

0 .0

5 .0

1 0 .0

1 5 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

Linear Nonlinear Linear

Nonlinear Linear Nonlinear

Line of Best Fit

For linearly related scatter diagram

Line is line of best fit.

Use graphing calculator to find

Example.

(a) Problem: Use a graphing utility to find the line of best fit to the data in the last example.

Answer:

Line of Best Fit

Example. (cont.)

(b) Problem: Graph the line of best fit from the last example on the scatter diagram.

Answer:

Line of Best Fit

Example. (cont.)

(c) Problem: Determine the loan amount that an individual would qualify for if her income is $42,000.

Answer:

Direct Variation

Variation or proportionality.

y varies directly with x, or is directly proportional to x:

There is a nonzero number such that y = kx.

k is the constant of proportionality.

Direct Variation

Example. Suppose y varies directly with x. Suppose as well that y = 15 when x = 3.

(a) Problem: Find the constant of proportionality.

Answer:

(b) Problem: Find x when y = 124.53.

Answer:

Key Points

Linear Functions


Scatter Diagrams

Linear and Nonlinear Relationships

Line of Best Fit

Direct Variation

Library of Functions;Piecewise-defined Functions

Section 2.5

Linear Functions

f(x) = mx+b, m and b a real number

Domain: (—∞, ∞)

Range: (—∞, ∞)

unless m = 0

Increasing on (—∞, ∞)

(if m > 0)

Decreasing on (—∞, ∞)

(if m < 0)

Constant on (—∞, ∞)

(if m = 0)

Constant Function

f(x) = b, b a real number

Special linear functions


Range: {b}

Even/odd/neither: Even (also odd if b = 0)

Constant on (—∞, ∞)

x-intercepts: None (unless b = 0)

y-intercept: y = b.

Identity Function

f(x) = xSpecial linear function


Range: (—∞, ∞)

Even/odd/neither: Odd


x-intercepts: x = 0

y-intercept: y = 0.

Square Function

f(x) = x2


Range: [0, ∞)

Even/odd/neither: Even

Increasing on (0, ∞)

Decreasing on (—∞, 0)

x-intercepts: x = 0

y-intercept: y = 0.

Cube Function

f(x) = x3


Range: (—∞, ∞)



x-intercepts: x = 0

y-intercept: y = 0.

Square Root Function

Domain: [0, ∞)

Range: [0, ∞)

Even/odd/neither: Neither


x-intercepts: x = 0

y-intercept: y = 0

Cube Root Function


Range: (—∞, ∞)



x-intercepts: x = 0

y-intercept: y = 0

Reciprocal Function

Domain: x ≠ 0

Range: x ≠ 0


Decreasing on (—∞, 0) ∪ (0, ∞)

x-intercepts: None

y-intercept: None

Absolute Value Function

f(x) = |x|Domain: (—∞, ∞)

Range: [0, ∞)

Even/odd/neither: Even


Decreasing on (—∞, 0)

x-intercepts: x = 0

y-intercept: y = 0


Can also write the absolute value function as

This is a piecewise-defined function.

Greatest Integer Function

f(x) = int(x)greatest integer less than or equal to x


Range: Integers (Z)

Even/odd/neither: Neither

y-intercept: y = 0

Called a step function

-7.5 -5 -2.5 2.5 5 7.5

-8

-6

-4

-2

2

4

6

8

Piecewise-defined Functions

Example. We can define a function differently on different parts of its domain.(a) Problem: Find f(—2)

Answer:

(b) Problem: Find f(—1)

Answer:

(c) Problem: Find f(2)

Answer:

(d) Problem: Find f(3)

Answer:

Key Points

Linear Functions

Constant Function

Identity Function

Square Function

Cube Function

Square Root Function

Cube Root Function

Reciprocal Function


Key Points (cont.)


Piecewise-defined Functions

Graphing Techniques: Transformations

Section 2.6

Transformations

Use basic library of functions and transformations to plot many other functions.

Plot graphs that look “almost” like one of the basic functions.

Shifts

Example.

Problem: Plot f(x) = x3, g(x) = x3 — 1 and h(x) = x3 + 2 on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Vertical shift:

A real number k is added to the right side of a function y = f(x),

New function y = f(x) + k

Graph of new function:

Graph of f shifted vertically up k units (if k > 0)

Down |k| units (if k < 0)

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Example.

Problem: Use the graph of f(x) = |x|to obtain the graph of g(x) = |x| + 2

Answer:

Shifts

Example.

Problem: Plot f(x) = x3, g(x) = (x — 1)3

and h(x) = (x + 2)3 on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Horizontal shift:

Argument x of a function f is replaced by x — h,

New function y = f(x — h)


Graph of f shifted horizontally right hunits (if h > 0)

Left |h| units (if h < 0)

Also y = f(x + h) in latter case

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Example.

Problem: Use the graph of f(x) = |x|to obtain the graph of g(x) = |x+2|

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Example.

Problem: The graph of a function y = f(x) is given. Use it to plot g(x) = f(x — 3) + 2

Answer:

Compressions and Stretches

Example.

Problem: Plot f(x) = x3, g(x) = 2x3 and on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Vertical compression/stretch:

Right side of function y = f(x) is multiplied by a positive number a,

New function y = af(x)


Multiply each y-coordinate on the graph of y = f(x) by a.

Vertically compressed (if 0 < a < 1)

Vertically stretched (if a > 1)


Example.

Problem: Use the graph of f(x) = x2

to obtain the graph of g(x) = 3x2

Answer: -4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Example.

Problem: Plot f(x) = x3, g(x) = (2x)3

and on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Horizontal compression/stretch:

Argument x of a function y = f(x) is multiplied by a positive number a

New function y = f(ax)


Divide each x-coordinate on the graph of y = f(x) by a.

Horizontally compressed (if a > 1)

Horizontally stretched (if 0 < a < 1)

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Example.

Problem: Use the graph of f(x) = x2

to obtain the graph of g(x) = (3x)2

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Example.

Problem: The graph of a function y = f(x) is given. Use it to plot g(x) = 3f(2x)

Answer:

Reflections

Example.

Problem: f(x) = x3 + 1 and

g(x) = —(x3 + 1) on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Reflections

Reflections about x-axis :

Right side of the function y = f(x) is multiplied by —1,

New function y = —f(x)


Reflection about the x-axis of the graph of the function y = f(x).

Reflections

Example.

Problem: f(x) = x3 + 1 and

g(x) = (—x)3 + 1 on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Reflections

Reflections about y-axis :

Argument of the function y = f(x) is multiplied by —1,

New function y = f(—x)


Reflection about the y-axis of the graph of the function y = f(x).

Summary of Transformations

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Example.

Problem: Use transformations to graph the function

Answer:

Key Points

Transformations

Shifts


Reflections


Mathematical Models: Constructing Functions

Section 2.7

Mathematical Models

Example.

Problem: The volume V of a right circular

cylinder is V = πr2h. If the height is

three times the radius, express the

volume V as a function of r.

Answer:

Mathematical Models

Example. Anne has 5000 feet of fencing available to enclose a rectangular field. One side of the field lies along a river, so only three sides require fencing.(a) Problem: Express the area A of the

rectangle as a function of x, where x is the length of the side parallel to the river.

Answer:

1000 2000 3000 4000 5000 6000

500000

1×106

1.5×106

2×106

2.5×106

3×106

3.5×106

Mathematical Models

Example (cont.)

(b) Problem: Graph A = A(x) and find what value of xmakes the area largest.

Answer:

(c) Problem: What value of x makes the area largest?

Answer:

Key Points

Mathematical Models

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