Lecture 1

1. Functions and Graphs

Pre Calculus Algebra and Trigonometry (Math 1150)Dawit G. TadesseAuburn University

Summer 2011

Pre Calculus Algebra and Trigonometry (Math 1150) Dawit G. Tadesse Auburn University1. Functions and Graphs

1.1 Graphs and Graphing utilities

Points and Ordered Pairs

The rectangular coordinate system or Cartesian coordinatesystem uses two number lines. The horizontal number line is thex-axis. The vertical number line is the y-axis. The point ofintersection of these axes is their zero points, called the origin.Positive numbers are shown to the right and above the origin.Negative numbers are shown to the left and below the origin. Theaxes divide the plane into four quarters, called quadrants. Thepoints located on the axes are not in any quadrant.

Each point in the rectangular coordinate system corresponds to anordered pair of real numbers, (x , y). The first number in eachpair, called the x-coordinate, denotes the distance and directionfrom the origin along the x-axis. The second number in each pair,called the y-coordinate, denotes vertical distance and directionalong a line parallel to the y-axis or along the y-axis itself.


1.1 Graphs and Graphing utilities(Contd.)

Figure: The rectangular coordinate system



Example 1: Plot the points A(2, 5), B(-1, 4), C(-4, -2), D(3, -2)in the rectangular coordinate system.

Graphs of Equations

A relationship between two quantities can be expressed as anequation in two variables. A solution of an equation in twovariables, x and y, is an ordered pair of real numbers. The graphof an equation in two variables is the set of all points whosecoordinates satisfy the equation. One method for graphing suchequations is the point-plotting method.

Example 2: Graph the following equations setting x= -3, -2, -1, 0,1, 2, and 3.a. y = x2 + 2b. y = x + 2c. y = −2|x |



InterceptsAn x-intercept of a graph is the x-coordinate of a point where thegraph intersects the x-axis. It is calculated by setting y=0. Ay-intercept of a graph is the y-coordinate of a point where thegraph intersects the y-axis. It is calculated by setting x=0.

Example 3: Find the x and y-intercepts for the graphs in Example2.

Home Work: Section 1.1 on page 143 problems 1, 5, 11, 13, 15,21, 41, 43.


1.2 Basics of Functions and Their Graphs

Relations

A relation is any set of ordered pairs. The set of all firstcomponents of the ordered pairs is called the domain of therelation and the set of all second components is called the rangeof the relation.

Example 1: Find the domain and range of the following relations:a. {(1, 2), (−1, 3), (0,−2)}b. {(−7,−7), (−5,−5), (0,−3), (2, 3)}

Functions

A relation in which each member of the domain corresponds toexactly one member of the range is a function.A function is a correspondence from a first set, called the domain,to a second set, called the range, such that each element in thedomain corresponds to exactly one element in the range.


1.2 Basics of Functions and Their Graphs

Example 2: Determine whether each relation is a function.a. {(10, 4), (−2, 4), (−1, 1), (5, 6)}b. {(5, 6), (5, 7), (6, 6), (6, 7)}

Functions as EquationsFunctions are usually given in terms of equations rather than assets of ordered pairs. For example, y = 3x + 1 represents afunction. For each value of x, there is one and only one value of y.The variable x is called the independent variable and the variabley is called the dependent variable.

Example 3: Determine whether an equation represents a function:a. x2 + y = 25b. 4x = y2

c. y = −√

x + 4


1.2 Basics of Functions and Their Graphs(contd.)

Function NotationIf an equation in x and y gives one and only one value of y foreach value of x , then the variable y is a function of x . When anequation represents a function, the function is often named by aletter such as f , g , h,F ,G , or H. Think of the domain as the set ofthe function’s inputs and the range as the set of the function’soutputs. Input can be represented by x and the output by f (x).The special notation f(x), read ”f of x” or ”f at x,” represents thevalue of the function at the number x. For example, theequation y = 3x + 1 defines y as a function of x. We can denotethe above equation as f (x) = 3x + 1.

Example 4: Let f (x) = 3x + 7. Evaluate the following:a. f (4) b. f (x + 1) c. f (−x)

Graphs of FunctionsThe graph of a function is the graph of its ordered pairs. Forexample, the graph of f (x) = 2x is the set of points (x , y) in



the rectangular coordinate system satisfying y = 2x .

Example 5: Graph the functions f (x) = x and g(x) = x − 4 inthe same coordinate system by letting x = −2,−1, 0, 1, 2.

The Vertical Line TestIf any vertical line intersects a graph in more than one point, thegraph does not define y as a function of x.

Example 6: Use the vertical line test to identify graphs in which yis a function of x.

Figure: aPre Calculus Algebra and Trigonometry (Math 1150) Dawit G. Tadesse Auburn University1. Functions and Graphs


Figure: b



Obtaining Information from GraphsYou can obtain information about a function from its graph.• A closed dot at the right or left of a graph indicates that thegraph does not extend beyond this point and the point belongs tothe graph.• An open dot at the right or left of a graph indicates that thegraph does not extend beyond this point and the point does notbelong to the graph.• An arrow indicates that the graph extends indefinitely in thedirection in which the arrow points.

Identifying Domain, Range and Intercepts from a Function’sGraphExample 7: Exercise set 1.2 on page 161 problems 78, 82.Home Work: Section 1.2 on page 159 problems 1, 3, 11, 13, 21,33, 43, 59, 77, 81.


1.3 More on Functions and Graphs

Increasing, Decreasing, and Constant Functions1. A function is increasing on an open interval, I, if f (x1) < f (x2)whenever x1 < x2 for any x1 and x2 in the interval.2. A function is decreasing on an open interval, I, if f (x1) > f (x2)whenever x1 < x2 for any x1 and x2 in the interval.3. A function is constant on an open interval, I, if f (x1) = f (x2)for any x1 and x2 in the interval.


1.3 More on Functions and Graphs(contd.)



Example 1: Use the graph to determine the intervals on which thefunction is increasing, decreasing or constant, if any.



Relative Maxima and Relative Minima1. A function value f (a) is a relative maximum (or localmaximum) of f if there exists an open interval containing a suchthat f (a) > f (x) for all x 6= a in the open interval.2. A function value f (b) is a relative minimum (or localminimum) of f if there exists an open interval containing b suchthat f (b) < f (x) for all x 6= b in the open interval.

Figure: -3.5, 2.5 are relative minimum points but -0.5 is relative min ptPre Calculus Algebra and Trigonometry (Math 1150) Dawit G. Tadesse Auburn University1. Functions and Graphs


Even and Odd Functions and SymmetryThe function f is an even function if

f (−x) = f (x) for all x in the domain of f .

The right side of the equation of an even function does not changeif x is replaced with -x.The function f is an odd function if

f (−x) = −f (x) for all x in the domain of f .

Every term on the right side of the equation of an odd functionchanges its sign if x is replaced with -x.Note that the graph of an even function in which f(-x)=f(x) issymmetric with respect to the y-axis.The graph of an odd function in which f (−x) = −f (x) issymmetric with respect to the origin.

Example 2: Determine whether each function is even, odd, orneither.



a. f (x) = 2x2 + x4

b. g(x) = x3 − xc. h(x) = x2 − x

Piecewise FunctionsA function that is defined by two (or more) equations over aspecified domain is called a piecewise function.

Example 3: Draw the graph of the piecewise function given by

f (x) =

{x if x < 0,

−x if x ≥ 0

Evaluate f (−2), f (0), f (2).

Functions and Difference QuotientsThe expression

f (x + h)− f (x)

h



for h 6= 0 is called the difference quotient of the function f.

Example 4: Find and simplify the difference quotient for the givenfunctiona.f (x) = 7xb. f (x) = x2 − 5x + 8

Home Work: Section 1.3 on page 172 problems 1, 9, 17, 19, 31,33, 37, 47, 57, 61.


1.4 Linear Functions and Slope

The Slope of a LineThe slope of the line through the distinct points(x1, y1) and (x2, y2) is

Change in y

Change in x=

Rise

Run=

Vertical Change

Horizontal Change

=y2 − y1

x2 − x1

where x2 − x1 6= 0.Notation: It is common to use the letter m to denote the slope ofa line.


1.4 Linear Functions and Slope(contd.)

Question: Draw the four possibilities for a Line’s Slope.Example 1: Find the slope of a line passing through the points:a. (2, 1) and (3, 4)b. (−1, 3) and (2, 4)

The Point-Slope Form of the Equation of a LineThe point-slope form of the equation of a non vertical(i.e. theslope is not undefined) line with slope m that passes through thepoint (x1, y1) is

y − y1 = m(x − x1)

Example 2: Write an equation in point-slope form for the linewith slope 4 that passes through the point (1, 3). Then solve theequation for y.

Example 3: Write an equation in point-slope form for the linepassing through the points (3, 5) and (8, 15).



Slope-Intercept Form of the Equation of a LineThe slope-intercept form of the equation of a non vertical line withslope m and y-intercept b is

y = mx + b.

The slope-intercept form of a line’s equation, y=mx+b, can beexpressed in function notation by replacing y with f(x):

f (x) = mx + b.

Such functions are called linear functions. Thus, in the equationof a linear function, the coefficient of x is the line’s slope and theconstant term is the y-intercept.

Example 4: Write an equation in slope-intercept form for a linewith x-intercept 4 and y-intercept -2.

Example 5: Give the slope and y-intercept of the line withequation f (x) = 3

4x − 3. Then graph the linear function.



Equation of a Horizontal LineA horizontal line is given by an equation of the form

y = b,

where b is the y-intercept of the line. The slope of a horizontal lineis zero.

Equation of a Vertical LineA vertical line is given by an equation of the form

x = a,

where a is the x-intercept of the line. The slope of a vertical line isundefined.

Example 6: Graph the equations x = 5 and y = 4.



The General Form of the Equation of a LineEvery line has an equation that can be written in the general form

Ax + By + C = 0,

where A,B, and C are real numbers, and A and B are not bothzero.

Example 7: Find the slope and the y-intercept of the line whoseequation is 4x + y − 6 = 0.

Example 8: Use intercepts to graph the equation3x + 5y + 15 = 0.

Home Work: Section 1.4 on page 188 problems 1, 11, 25, 37, 43,51, 59, 67.


1.5 More on Slope

Parallel and Perpendicular LinesTwo non intersecting lines that lie in the same plane are parallel.Properties:1. If two non vertical lines are parallel, then they have the sameslope.2. If two distinct non vertical lines have the same slope, then theyare parallel.3. Two distinct vertical lines, both with undefined slopes, areparallel.

Example 1: Write an equation of the line passing through(−2,−7) and parallel to the line whose equation is y = 5x − 4.

Two lines that intersect at a right angle (90◦) are said to beperpendicular.Properties1. If two non vertical lines are perpendicular, then the product oftheir slopes is -1.


1.5 More on Slope(contd.)

2. If the product of the slopes of two lines is -1, then the lines areperpendicular.3. A horizontal line having zero slope is perpendicular to a verticalline having undefined slope.

Example 2: Write an equation of the line passing through (−4, 2)and perpendicular to the line whose equation is y = 1

3x + 7.

Slope as Rate of ChangeSlope is defined as the ratio of a change in y to a correspondingchange in x. It describes how fast y is changing with respect to x.For a linear function, slope may be interpreted as the rate ofchange of the dependent variable per unit change in theindependent variable.

Example 3: Let y and x are related by the equation−2x + 3y + 3 = 0. Find the rate of change of y with respect to x.



The Average Rate of Change of a FunctionIf the graph of a function is not a straight line, the average rateof change between any two points is the slope of the linecontaining the two pints. This line is called a secant line.Let (x1, f (x1)) and (x2, f (x2)) be distinct points on the graph of afunction f. The average rate of change of f from x1 to x2,denoted by ∆y

∆x (read ”delta y divided by delta x” or ”change in ydivided by change in x”), is

∆y

∆x=

f (x2)− f (x1)

x2 − x1.

Example 4: Find the average rate of change of the functionf (x) = x2 − 2x from:a. x1 = 0 to x2 = 4b. x1 = 3 to x2 = 6



Average Velocity of an ObjectSuppose that a function expresses an object’s position, s(t), interms of time, t. The average velocity of the object from t1 to t2is

∆s

∆t=

s(t2)− s(t1)

t2 − t1.

Example 5: Suppose that a ball is rolling down a ramp. Thedistance traveled by the ball is given by s(t) = 12t2, where t is thetime, in seconds, after the ball is released and s(t) is measured infeet. Find the ball’s average velocity froma. t1 = 3 to t2 = 4.b. t1 = 3 to t2 = 3.5.

Home Work: Section 1.5 on page 200 problems 3, 5, 7, 11, 13,15, 19.


1.6 Transformations of Functions

Graphs of Common FunctionsThe following are graphs of some of the common functions inAlgebra:Constant Function

Figure: Graph of f(x)=c


1.6 Transformations of Functions(Contd.)

Identity Function

Figure: Graph of f(x)=x

Absolute Value Function

Figure: Graph of f (x) = |x |Pre Calculus Algebra and Trigonometry (Math 1150) Dawit G. Tadesse Auburn University1. Functions and Graphs


Standard Quadratic Function

Figure: Graph of f (x) = x2

Square Root Function

Figure: Graph of f (x) =√

xPre Calculus Algebra and Trigonometry (Math 1150) Dawit G. Tadesse Auburn University1. Functions and Graphs


Standard Cubic Function

Figure: Graph of f (x) = x3

Cubic Root Function

Figure: Graph of f (x) = 3√

x



Vertical ShiftsLet f be a function and c a positive real number.• The Graph of y = f (x) + c is the graph of y = f (x) shifted cunits vertically upward.• The Graph of y = f (x)− c is the graph of y = f (x) shifted cunits vertically downward.

Example 1: Use the graph of f (x) = x2 to obtain the graphs ofg(x) = x2 + 1 and h(x) = x2 − 1.

Horizontal ShiftsLet f be a function and c a positive real number.• The graph of y = f (x + c) is the graph of y = f (x) shifted tothe left c units.• The graph of y = f (x − c) is the graph of y = f (x) shifted tothe right c units.

Example 2: Use the graph of f (x) = x2 to obtain the graphs ofg(x) = (x + 1)2 and h(x) = (x − 1)2.



Example 3: Use the graph of f (x) = x2 to obtain the graphs ofg(x) = (x + 1)2 + 1 and h(x) = (x − 1)2 − 1.

Reflections of Graphs

Reflection about the x-AxisThe graph of y = −f (x) is the graph of y = f (x) reflected aboutthe x-axis.Example 4: Use the graph of f (x) =

√x to obtain the graph of

g(x) = −√

x .

Reflection about the y-AxisThe graph of y = f (−x) is the graph of y = f (x) reflected aboutthe y-axis.Example 6: Use the graph of f (x) =

√x to obtain the graph of

g(x) =√−x .

Vertically Stretching and Shrinking GraphsLet f be a function and c a positive real number.



• If c > 1, the graph of y = cf (x) is the graph of y = f (x)vertically stretched by multiplying each of its y-coordinates by c.• If 0 < c < 1, the graph of y = cf (x) is the graph of y = f (x)vertically shrunk by multiplying each of its y-coordinates by c.

Example 7: Use the graph of f (x) = |x | to obtain the graphs ofg(x) = 2|x | and h(x) = 1

2 |x |.

Horizontally Stretching and Shrinking GraphsLet f be a function and c a positive real number.• If c > 1, the graph of y = f (cx) is the graph of y = f (x)horizontally shrunk by dividing each of its x-coordinates by c.• If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x)horizontally stretched by dividing each of its x-coordinates by c.

Example 8: Use the graph of f (x) = |x | to obtain the graphs ofg(x) = |2x | and h(x) = |12x |.



Sequences of TransformationsA function involving more than one transformation can be graphedby performing transformations in the following order:

1. Horizontal shifting 2. Stretching or shrinking3. Reflecting 4. Vertical shifting

Example 9: Use the graph of f (x) = x3 to obtain the graph ofg(x) = 1

2(x − 2)3 − 1.

Home Work: Section 1.6 on page 216 problems 1, 17, 33, 57, 59,75, 87, 93, 105, 117.


1.7 Combinations of Functions; Composite Functions

Finding a Function’s DomainThe domain of a function f is the largest set of real numbers forwhich the value of f (x) is a real number. Exclude from afunction’s domain real numbers that cause division by zero andreal numbers that result in an even root a negative number.

Example 1: Find the domain of each function.

a. f (x) = 2(x + 5)

b. g(x) = 2x+5

c. g(x) = 2x2+x−12

d. f (x) =√

x + 2

The Algebra of Functions: Sum, Difference, Product, andQuotient of FunctionsLet f and g be two functions. The sum f + g , the differencef − g , the product fg , and the quotient f

g are functions whose


1.7 Combinations of Functions; CompositeFunctions(contd.)

domains are the set of all real numbers common to the domains off and g(Df ∩ Dg ), defined as follows:

1. Sum: (f + g)(x) = f (x) + g(x)

2. Difference: (f − g)(x) = f (x)− g(x)

3. Product: (fg)(x) = f (x).g(x)

4. Quotient: ( fg )(x) = f (x)

g(x) , provided g(x) 6= 0.

Example 2: Let f (x) =√

x + 6 and g(x) =√

x − 3. Find each ofthe following functions:

a. (f + g)(x) b. (f − g)(x) c. (fg)(x) d. ( fg )(x).

Determine the domain for each function.

Composite FunctionsThe composition of FunctionsThe composition of the function f with g is denoted by



f ◦ g and is defined by the equation

(f ◦ g)(x) = f (g(x)).

The domain of the composite function f ◦ g is the set of all xsuch that1. x is in the domain of g and2. g(x) is in the domain of f .

Excluding Values from the domain of (f ◦ g)(x) = f (g(x)) Thefollowing values must be excluded from the input x :• If x is not in the domain of g , it must not be in the domain off ◦ g .• Any x for which g(x) is not in the domain of f must not be inthe domain of f ◦ g .

Example 3: Given f (x) = 5x+4 , g(x) = 1

x . Find



a. (f ◦ g)(x) b. (g ◦ f )(x) c. (f ◦ g)(−1) d. Domainof f ◦ g .

Decomposing Functions

Example 4: Express the function h(x) = (2x − 5)3 as acomposition of two functions f and g so that h(x) = (f ◦ g)(x).

Home Work: Section 1.7 on page 229 problems 1,3, 13, 19, 31,39, 49, 55, 65, 79.


1.8 Inverse Functions

Definition of the Inverse of a FunctionLet f and g be two functions such that

f (g(x)) = x for every x in the domain of g

andg(f (x)) = x for every x in the domain of f .

The function g is the inverse of the function f and is denoted byf −1 (read ”f -inverse”). Thus, f (f −1(x)) = x and f −1(f (x)) = x .The domain of f is equal to the range of f −1, and vice versa.

Example 1: Show that each function is the inverse of the other:a. f (x) = 4x + 9 and g(x) = x−9

4

b. f (x) = 2x−5 and g(x) = 2

x + 5.

Finding the Inverse of a FunctionThe equation for the inverse of a function f can be found asfollows:


1.8 Inverse Functions(Cont.)

1. Replace f (x) with y in the equation for f (x).

2. Interchange x and y .

3. Solve for y . If this equation does not define y as a function ofx , the function f does not have an inverse function and thisprocedure ends. If this equation does define y as a function of x ,the function f has an inverse function.

4. If f has an inverse function, replace y in step 3 by f −1(x). Wecan verify our result by showing that f (f −1(x)) = x andf −1(f (x)) = x .

Example 2: Find the inverse of each function:a. f (x) = 3x − 1

b. f (x) = 2x−3x+1 .

The Horizontal Line Test for Inverse FunctionsA function f has an inverse that is a function, f −1, if there is nohorizontal line that intersects the graph of the function f at morethan one point.



Figure: Have an Inverse

Figure: Have NO InversePre Calculus Algebra and Trigonometry (Math 1150) Dawit G. Tadesse Auburn University1. Functions and Graphs


One-to-One FunctionA one-to-one function is a function in which no two differentordered pairs have the same second component. Only one-to-onefunctions have inverse functions. For example, f (x) = x2 has noinverse because (−2, 4) and (2, 4) have the same secondcomponent, that is, 4. Hence f (x) = x2 have no inverse.

Graphs of f and f −1

The graph of f −1 is a reflection of the graph of f about the liney = x .



Finding the Inverse of a Domain-Restricted FunctionExample 3: Find an equation for f −1(x) of each function:a. f (x) = x2 − 1, x ≤ 0

b. f (x) = (x − 1)2, x ≥ 1.

Home Work: Section 1.8 on page 240 problems 3, 7, 25, 27, 29,33, 35, 41, 43. 33,


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