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Chapter 3: Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 3: Functions

346 problems · 130 figures

Try It #1 §3.1 source ↗

Table 21 lists the five greatest baseball players of all time in order of rank.

Player Rank
Babe Ruth 1
Willie Mays 2
Ty Cobb 3
Walter Johnson 4
Hank Aaron 5
Table 2

ⓐIs the rank a function of the player name?

ⓑIs the player name a function of the rank?

Answer

ⓐyes

ⓑ yes (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

Try It #2 §3.1 source ↗

Use function notation to express the weight of a pig in pounds as a function of its age in days d.

Answer

w=f(d)

Try It #3 §3.1 source ↗

Does Table 9 represent a function?

Input Output
1 10
2 100
3 1000
Table 9

Answer

yes

Try It #4 §3.1 source ↗

Given the function g( m )= m−4 , evaluate g( 5 ).

Answer

g( 5 )=1

Try It #5 §3.1 source ↗

Given the function g( m )= m−4 , solve g( m )=2.

Answer

m=8

Try It #6 §3.1 source ↗

If x−8 y 3 =0, express y as a function of x.

Answer

y=f( x )= x 3 2

Try It #7 §3.1 source ↗

Using the table from Evaluating and Solving a Tabular Function above, evaluate g( 1 ).

Answer

g( 1 )=8

Try It #8 §3.1 source ↗

Using Figure 4, solve f( x )=1.

Answer

x=0 or x=2

Try It #9 §3.1 source ↗

ⓐ Is a balance a function of the bank account number?

ⓑ Is a bank account number a function of the balance?

ⓒ Is a balance a one-to-one function of the bank account number?

Answer

ⓐ yes, because each bank account has a single balance at any given time;

ⓑ no, because several bank account numbers may have the same balance;

ⓒ no, because the same output may correspond to more than one input.

Try It #10 §3.1 source ↗

Evaluate the following:
ⓐ If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
ⓑ If so, is the function one-to-one?

Answer

ⓐ Yes, letter grade is a function of percent grade;

ⓑ No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

Try It #11 §3.1 source ↗

Does the graph in Figure 11 represent a function?

Graph of absolute value function. Figure 11

Answer

yes

Try It #12 §3.1 source ↗

Is the graph shown in Figure 9(c) one-to-one?

Answer

No, because it does not pass the horizontal line test.

3.1-1 §3.1 source ↗

What is the difference between a relation and a function?

Answer

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

3.1-3 §3.1 source ↗

Why does the vertical line test tell us whether the graph of a relation represents a function?

Answer

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

3.1-5 §3.1 source ↗

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

Answer

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

3.1-7 §3.1 source ↗

For the following exercises, determine whether the relation represents a function.

{(a,b),(b,c),(c,c)}

Answer

function

3.1-9 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

y= x 2

Answer

function

3.1-11 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

3 x 2 +y=14

Answer

function

3.1-13 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

y=−2 x 2 +40x

Answer

function

3.1-15 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

x= 3y+5 7y−1

Answer

function

3.1-17 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

y= 3x+5 7x−1

Answer

function

3.1-19 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

2xy=1

Answer

function

3.1-21 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

y= x 3

Answer

function

3.1-23 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

x=± 1−y

Answer

function

3.1-25 §3.1 source ↗

For the following exercises, determine whether the relation represents y as a function of x.

y 2 = x 2

Answer

not a function

3.1-27 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

f(x)=2x−5

Answer

f(−3)=−11;
f(2)=−1;
f(−a)= − 2a− 5;
− f(a)= − 2a+ 5;
f(a+h)= 2a+ 2h− 5

3.1-29 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

f(x)= 2−x +5

Answer

f(−3)= 5 +5;
f(2)=5;
f(−a)= 2+a + 5;
− f(a)= − 2−a − 5;
f(a+h)= 2−a−h + 5

3.1-31 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

f(x)= |x− 1|− |x+ 1|

Answer

f(−3)=2; f(2)= 1− 3= − 2;
f(−a)= |− a− 1|− |− a+ 1|;
− f(a)= − |a− 1|+ |a+ 1|;
f(a+h)= |a+ h− 1|− |a+ h+ 1|

3.1-33 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

Given the function g(x)= x 2 +2x, evaluate g(x)−g(a) x−a , x≠a.

Answer

g(x)−g(a) x−a = x+ a+ 2, x≠a

3.1-35 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

Given the function f(x)=8−3x:

ⓐ Evaluate f(−2).

ⓑ Solve f(x)=−1.

Answer

a. f(−2)=14; b. x=3

3.1-37 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

Given the function f(x)= x 2 −3x:

ⓐ Evaluate f(5).

ⓑ Solve f(x)=4.

Answer

a. f(5)=10; b. x=−1 or x=4

3.1-39 §3.1 source ↗

For the following exercises, evaluate f(−3), f(2), f(−a), −f(a), f(a+h).

Consider the relationship 3r+2t=18.

ⓐ Write the relationship as a function r=f(t).

ⓑ Evaluate f(−3).

ⓒ Solve f(t)=2.

Answer

ⓐ f(t)=6− 2 3 t;

ⓑ f(−3)=8;

ⓒ t=6

3.1-41 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

Graph of x equals y squared plus one, showing a sideways parabola opening to the right from the vertex (1,0).

Answer

not a function

3.1-43 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

A graph displays a piecewise linear function. It's a horizontal line at y=-1 for x less than -1, linearly increases from (-1,-1) to (1,1), then becomes a horizontal line at y=1 for x greater than 1.

Answer

function

3.1-45 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

Graph of relation.

Answer

function

3.1-47 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

A linear graph displaying a straight line with a negative slope, intersecting the x-axis at 2 and the y-axis at 2.

Answer

function

3.1-49 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

A graph displaying a straight line with a positive slope passing through the origin in a Cartesian coordinate system.

Answer

function

3.1-51 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

A graph on an x-y coordinate plane shows a sinusoidal wave centered on the x-axis. The wave passes through the origin (0,0) and oscillates, extending indefinitely with arrows on both ends.

Answer

function

3.1-53 §3.1 source ↗

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

Given the following graph,

ⓐ Evaluate f(0).

ⓑ Solve for f(x)=−3.

A graph of a W-shaped quartic function on a Cartesian coordinate plane. The curve shows two local minima and one local maximum, intersecting the x-axis at four points.

Answer

ⓐ f(0)=1;

ⓑ f(x)= − 3, x= − 2 or x=2

3.1-55 §3.1 source ↗

For the following exercises, determine if the given graph is a one-to-one function.

Graph of a circle.

Answer

not a function so it is also not a one-to-one function

3.1-57 §3.1 source ↗

For the following exercises, determine if the given graph is a one-to-one function.

Graph of a rotated cubic function.

Answer

one-to- one function

3.1-59 §3.1 source ↗

For the following exercises, determine if the given graph is a one-to-one function.

Graph of a one-to-one function.

Answer

function, but not one-to-one

3.1-61 §3.1 source ↗

For the following exercises, determine whether the relation represents a function.

{ ( 3,4 ),( 4,5 ),( 5,6 ) }

Answer

function

3.1-63 §3.1 source ↗

For the following exercises, determine if the relation represented in table form represents y as a function of x.

x 5 10 15
y 3 8 14

Answer

function

3.1-65 §3.1 source ↗

For the following exercises, determine if the relation represented in table form represents y as a function of x.

x 5 10 10
y 3 8 14

Answer

not a function

3.1-67 §3.1 source ↗

For the following exercises, use the function f represented in the table below.

x 0 1 2 3 4 5 6 7 8 9
f(x) 74 28 1 53 56 3 36 45 14 47
Table 14

Solve f(x)=1.

Answer

f(x)=1,x=2

3.1-69 §3.1 source ↗

For the following exercises, evaluate the function f at the values f(−2), f(−1), f(0), f(1), and f(2).

f( x )=8−3x

Answer

f(−2)=14; f(−1)=11; f(0)=8; f(1)=5; f(2)=2

3.1-71 §3.1 source ↗

For the following exercises, evaluate the function f at the values f(−2), f(−1), f(0), f(1), and f(2).

f( x )=3+ x+3

Answer

f(−2)=4; f(−1)=4.414; f(0)=4.732; f(1)=5; f(2)=5.236

3.1-73 §3.1 source ↗

For the following exercises, evaluate the function f at the values f(−2), f(−1), f(0), f(1), and f(2).

f( x )= 3 x

Answer

f(−2)= 1 9 ; f(−1)= 1 3 ; f(0)=1; f(1)=3; f(2)=9

3.1-75 §3.1 source ↗

For the following exercises, evaluate the expressions, given functions f,g, and h:

f(x)=3x−2
g(x)=5− x 2
h(x)= − 2 x 2 + 3x− 1

f( 7 3 )−h( −2 )

Answer

20

3.1-77 §3.1 source ↗

For the following exercises, graph y= x 2 on the given domain. Determine the corresponding range. Show each graph.

[−10,10]

Answer

[0,100]

Graph of a parabola.

3.1-79 §3.1 source ↗

For the following exercises, graph y= x 3 on the given domain. Determine the corresponding range. Show each graph.

[−0.1,0.1]

Answer

[−0.001,0.001]

Graph of a parabola.

3.1-81 §3.1 source ↗

For the following exercises, graph y= x 3 on the given domain. Determine the corresponding range. Show each graph.

[−100,100]

Answer

[−1,000,000,1,000,000]

Graph of a cubic function.

3.1-83 §3.1 source ↗

For the following exercises, graph y= x on the given domain. Determine the corresponding range. Show each graph.

[0,100]

Answer

[0,10]

Graph of a square root function.

3.1-85 §3.1 source ↗

For the following exercises, graph y= x 3 on the given domain. Determine the corresponding range. Show each graph.

[−0.001,0.001]

Answer

[−0.1,0.1]

Graph of a square root function.

3.1-87 §3.1 source ↗

For the following exercises, graph y= x 3 on the given domain. Determine the corresponding range. Show each graph.

[−1,000,000,1,000,000]

Answer

[−100,100]

Graph of a cubic root function.

3.1-89 §3.1 source ↗

For the following exercises, graph y= x 3 on the given domain. Determine the corresponding range. Show each graph.

The number of cubic yards of dirt, D, needed to cover a garden with area a square feet is given by D=g( a ).

ⓐ A garden with area 5000 ft2 requires 50 yd3 of dirt. Express this information in terms of the function g.

ⓑ Explain the meaning of the statement g( 100 )=1.

Answer

ⓐ g(5000)=50;

ⓑ The number of cubic yards of dirt required for a garden of 100 square feet is 1.

3.1-91 §3.1 source ↗

For the following exercises, graph y= x 3 on the given domain. Determine the corresponding range. Show each graph.

Let h( t ) be the height above ground, in feet, of a rocket t seconds after launching. Explain the meaning of each statement:

ⓐ h( 1 )=200

ⓑ h( 2 )=350

Answer

ⓐ The height of a rocket above ground after 1 second is 200 ft.

ⓑ The height of a rocket above ground after 2 seconds is 350 ft.

Try It #1 §3.2 source ↗

Find the domain of the function:

{ (−5,4),(0,0),(5,−4),(10,−8),(15,−12) }

Answer

{−5,0,5,10,15}

Try It #2 §3.2 source ↗

Find the domain of the function: f(x)= 5− x+ x 3 .

Answer

( −∞,∞ )

Try It #3 §3.2 source ↗

Find the domain of the function: f(x)= 1+4x 2x−1 .

Answer

( −∞, 1 2 )∪( 1 2 ,∞ )

Try It #4 §3.2 source ↗

Find the domain of the function f(x)= 5+2x .

Answer

[ − 5 2 ,∞ )

Try It #5 §3.2 source ↗

Given Figure 7, specify the graphed set in

ⓐ words

ⓑ set-builder notation

ⓒ interval notation

Line graph of -2≤x, -1≤x less than 3. Figure 7

Answer

ⓐ values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3

ⓑ { x|x≤−2or−1≤x<3 }

ⓒ (−∞,−2]∪[−1,3)

Try It #6 §3.2 source ↗

Given Figure 12, identify the domain and range using interval notation.

Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year. Figure 12

Answer

domain =[1950,2002] range = [47,000,000,89,000,000]

Try It #7 §3.2 source ↗

Find the domain and range of f( x )=− 2−x .

Answer

domain: ( −∞,2 ]; range: ( −∞,0 ]

Try It #8 §3.2 source ↗

Graph the following piecewise function.

f(x)={ x 3 if x<−1 −2 if −1<x<4 x if x>4

Answer

This graph shows a piecewise function with a curve extending left, a horizontal line segment, and a line segment extending right. Open and closed circles mark function behavior at specific points, indicating discontinuities.

3.2-1 §3.2 source ↗

Why does the domain differ for different functions?

Answer

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

3.2-3 §3.2 source ↗

Explain why the domain of f(x)= x 3 is different from the domain of f(x)= x .

Answer

There is no restriction on x for f(x)= x 3 because you can take the cube root of any real number. So the domain is all real numbers, (−∞,∞). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x -values are restricted for f(x)= x to nonnegative numbers and the domain is [0,∞).

3.2-5 §3.2 source ↗

How do you graph a piecewise function?

Answer

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x -axis and y -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate −∞ or ∞. Combine the graphs to find the graph of the piecewise function.

3.2-7 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)=5−2 x 2

Answer

(−∞,∞)

3.2-9 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f( x )=3− 6−2x

Answer

(−∞,3]

3.2-11 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= x 2 +4

Answer

(−∞,∞)

3.2-13 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= x−1 3

Answer

(−∞,∞)

3.2-15 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= 3x+1 4x+2

Answer

(−∞,− 1 2 )∪(− 1 2 ,∞)

3.2-17 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= x−3 x 2 +9x−22

Answer

(−∞,−11)∪(−11,2)∪(2,∞)

3.2-19 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= 2 x 3 −250 x 2 −2x−15

Answer

(−∞,−3)∪(−3,5)∪(5,∞)

3.2-21 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= 2x+1 5−x

Answer

(−∞,5)

3.2-23 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= x−6 x−4

Answer

[6,∞)

3.2-25 §3.2 source ↗

For the following exercises, find the domain of each function using interval notation.

f(x)= x 2 −9x x 2 −81

Answer

( −∞,−9 )∪( −9,9 )∪( 9,∞ )

3.2-27 §3.2 source ↗

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from (2, 8].

Answer

domain: (2,8], range [6,8)

3.2-29 §3.2 source ↗

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from [-4, 4].

Answer

domain: [−4,4], range: [0,2]

3.2-31 §3.2 source ↗

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from [-5, 3).

Answer

domain: [−5,3), range: [ 0,2 ]

3.2-33 §3.2 source ↗

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from (-infinity, 2].

Answer

domain: (−∞,1], range: [0,∞)

3.2-35 §3.2 source ↗

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from [-6, -1/6]U[1/6, 6]/.

Answer

domain: [ −6,− 1 6 ]∪[ 1 6 ,6 ]; range: [ −6,− 1 6 ]∪[ 1 6 ,6 ]

3.2-37 §3.2 source ↗

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from [-3, infinity).

Answer

domain: [−3,∞); range: [0,∞)

3.2-39 §3.2 source ↗

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

f(x)={ 2x−1 if x<1 1+x if x≥1

Answer

domain: (−∞,∞)

A graph displays a piecewise function with a jump discontinuity at x=1. The function is defined by a line segment ending with an open circle at (1,1) and another line segment starting with a closed circle at (1,2).

3.2-41 §3.2 source ↗

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

f( x )={ 3 if x<0 x if x≥0

Answer

domain: (−∞,∞)

Graph of f(x).

3.2-43 §3.2 source ↗

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

f(x)={ x 2 x+2 ifx<0 ifx≥0

Answer

domain: (−∞,∞)

A piecewise graph: a parabola curving left from an open circle at (0,0), and a line segment starting from a closed circle at (0,2) extending right. It illustrates a jump discontinuity.

3.2-45 §3.2 source ↗

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

f(x)={ |x| 1 ifx<2 ifx≥2

Answer

domain: (−∞,∞)

Graph of f(x).

3.2-47 §3.2 source ↗

For the following exercises, given each function f, evaluate f(−3), f(−2), f(−1), and f(0).

f(x)={ 1 if x≤−3 0 if x>−3

Answer

f(−3)=1; f(−2)=0; f(−1)=0; f(0)=0

3.2-49 §3.2 source ↗

For the following exercises, given each function f, evaluate f(−1), f(0), f(2), and f(4).

f(x)={ 7x+3 if x<0 7x+6 if x≥0

Answer

f(−1)=−4; f(0)=6; f(2)=20; f(4)=34

3.2-51 §3.2 source ↗

For the following exercises, given each function f, evaluate f(−1), f(0), f(2), and f(4).

f( x )={ 5x if x<0 3 if 0≤x≤3 x 2 if x>3

Answer

f(−1)=−5; f(0)=3; f(2)=3; f(4)=16

3.2-53 §3.2 source ↗

For the following exercises, write the domain for the piecewise function in interval notation.

f(x)={ x 2 −2ifx<1 − x 2 +2ifx>1

Answer

domain: (−∞,1)∪(1,∞)

3.2-55 §3.2 source ↗

For the following exercises, write the domain for the piecewise function in interval notation.

Graph y= 1 x 2 on the viewing window [−0.5,−0.1] and [0.1,0.5]. Determine the corresponding range for the viewing window. Show the graphs.

Answer

Graph of the equation from [-0.5, -0.1].

window: [−0.5,−0.1]; range: [4,100]

Graph of the equation from [0.1, 0.5].

window: [0.1,0.5]; range: [4,100]

3.2-57 §3.2 source ↗

For the following exercises, write the domain for the piecewise function in interval notation.

Suppose the range of a function f is [−5,8]. What is the range of |f(x)|?

Answer

[0,8]

3.2-59 §3.2 source ↗

For the following exercises, write the domain for the piecewise function in interval notation.

Create a function in which the domain is x>2.

Answer

Many answers. One function is f(x)= 1 x−2 .

3.2-61 §3.2 source ↗

For the following exercises, write the domain for the piecewise function in interval notation.

The cost in dollars of making x items is given by the function C(x)= 10x+ 500.

ⓐThe fixed cost is determined when zero items are produced. Find the fixed cost for this item.

ⓑWhat is the cost of making 25 items?

ⓒSuppose the maximum cost allowed is $1500. What are the domain and range of the cost function, C(x)?

Answer

ⓐ The fixed cost is $500.

ⓑ The cost of making 25 items is $750.

ⓒ The domain is [0, 100] and the range is [500, 1500].

Try It #1 §3.3 source ↗

Using the data in Table 1, find the average rate of change between 2005 and 2010.

Answer

$2.84−$2.31 5years = $0.53 5years = $0.106 per year.

Try It #2 §3.3 source ↗

Find the average rate of change of f( x )=x−2 x on the interval [1,9].

Answer

1 2

Try It #3 §3.3 source ↗

Find the average rate of change of f(x)= x 2 + 2x− 8 on the interval [5,a] in simplest forms in terms of a.

Answer

a+7

Try It #4 §3.3 source ↗

Graph the function f(x)= x 3 − 6 x 2 − 15x+ 20 to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.

Answer

The local maximum appears to occur at (−1,28), and the local minimum occurs at (5,−80). The function is increasing on (−∞,−1)∪(5,∞) and decreasing on (−1,5).

Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

3.3-1 §3.3 source ↗

Can the average rate of change of a function be constant?

Answer

Yes, the average rate of change of all linear functions is constant.

3.3-3 §3.3 source ↗

How are the absolute maximum and minimum similar to and different from the local extrema?

Answer

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

3.3-5 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

f( x )=4 x 2 −7 on [1,b]

Answer

4( b+1 )

3.3-7 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

p( x )=3x+4 on [2,2+h]

Answer

3

3.3-9 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

f( x )=2 x 2 +1 on [x,x+h]

Answer

4x+2h

3.3-11 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

a( t )= 1 t+4 on [9,9+h]

Answer

−1 13( 13+h )

3.3-13 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

j( x )=3 x 3 on [1,1+h]

Answer

3 h 2 +9h+9

3.3-15 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

Find f( x+h )−f( x ) h given f( x )=2 x 2 −3x on [x,x+h]

Answer

4x+2h−3

3.3-17 §3.3 source ↗

For the following exercises, consider the graph of f shown in Figure 15.

Graph of a polynomial. Figure 15

Estimate the average rate of change from x=2 to x=5.

Answer

4 3

3.3-19 §3.3 source ↗

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

Graph of a cubic function.

Answer

increasing on ( −∞,−2.5 )∪( 1,∞ ), decreasing on (−2.5,1)

3.3-21 §3.3 source ↗

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

Graph of a reciprocal function.

Answer

increasing on ( −∞,1 )∪( 3,4 ), decreasing on ( 1,3 )∪( 4,∞ )

3.3-23 §3.3 source ↗

For the following exercises, consider the graph shown in Figure 16.

Graph of a cubic function. Figure 16

Estimate the point(s) at which the graph of f has a local maximum or a local minimum.

Answer

local maximum: (−3,60), local minimum: (3,−60)

3.3-25 §3.3 source ↗

For the following exercises, consider the graph in Figure 17.

Graph of a cubic function. Figure 17

If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

Answer

absolute maximum at approximately (7,150), absolute minimum at approximately (−7.5,−220)

3.3-27 §3.3 source ↗

For the following exercises, consider the graph in Figure 17.

Graph of a cubic function. Figure 17

Table 4 gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Year Population
(thousands)
200087
200184
200283
200380
200477
200576
200678
200781
200885
Table 4

Answer

ⓐ –3000

ⓑ –1250

3.3-29 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified.

h( x )=5−2 x 2 on [−2,4]

Answer

-4

3.3-31 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified.

g( x )=3 x 3 −1 on [−3,3]

Answer

27

3.3-33 §3.3 source ↗

For the following exercises, find the average rate of change of each function on the interval specified.

p(t)= ( t 2 −4 )( t+1 ) t 2 +3 on [−3,1]

Answer

–0.167

3.3-35 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

f(x)= x 4 − 4 x 3 + 5

Answer

Local minimum at (3,−22), decreasing on (−∞,3), increasing on (3,∞)

3.3-37 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

g( t )=t t+3

Answer

Local minimum at (−2,−2), decreasing on (−3,−2), increasing on (−2,∞)

3.3-39 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

m(x)= x 4 + 2 x 3 − 12 x 2 − 10x+ 4

Answer

Local maximum at (−0.39,5.98), local minima at (−3.15,−47.62) and (2.04,-32.04), decreasing on (−∞,−3.15)∪ (−0.39,2.04), increasing on (−3.15,−0.39)∪ (2.04,∞)

3.3-41 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

The graph of the function f is shown in Figure 18.

Graph of f(x) on a graphing calculator. Figure 18

Based on the calculator screen shot, the point (1.333,5.185) is which of the following?

ⓐa relative (local) maximum of the function

ⓑthe vertex of the function

ⓒthe absolute maximum of the function

ⓓa zero of the function

Answer

A

3.3-43 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

Let f( x )= 1 x . Find the number b such that the average rate of change of f on the interval (2,b) is − 1 10 .

Answer

b=5

3.3-45 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

Answer

2.7 gallons per minute

3.3-47 §3.3 source ↗

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

The graph in Figure 19 illustrates the decay of a radioactive substance over t days.

Graph of an exponential function. Figure 19

Use the graph to estimate the average decay rate from t=5 to t=15.

Answer

approximately –0.6 milligrams per day

Try It #1 §3.4 source ↗

Find and simplify the functions ( fg )( x ) and ( f−g )( x ).

f( x )=x−1   and    g( x )= x 2 −1

Are they the same function?

Answer

( fg )( x )=f( x )g( x )=( x−1 )( x 2 −1 )= x 3 − x 2 −x+1 ( f−g )( x )=f( x )−g( x )=( x−1 )−( x 2 −1 )=x− x 2

No, the functions are not the same.

Try It #2 §3.4 source ↗

The gravitational force on a planet a distance r from the sun is given by the function G(r). The acceleration of a planet subjected to any force F is given by the function a(F). Form a meaningful composition of these two functions, and explain what it means.

Answer

A gravitational force is still a force, so a( G(r) ) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G( a(F) ) does not make sense.

Try It #3 §3.4 source ↗

Using Table 1, evaluate f(g(1)) and g(f(4)).

Answer

f(g(1))= f(3)= 3 and g(f(4))= g(1)= 3

Try It #4 §3.4 source ↗

Using Figure 1, evaluate g(f(2)).

Answer

g(f(2))= g(5)= 3

Try It #5 §3.4 source ↗

Given f(t)= t 2 −t and h(x)=3x+2, evaluate

ⓐ h(f(2))

ⓑ h(f(−2))

Answer

ⓐ 8

ⓑ 20

Try It #6 §3.4 source ↗

Find the domain of

( f∘g )(x) wheref(x)= 1 x−2  and  g(x)= x+4

Answer

[ −4,0 )∪( 0,∞ )

Try It #7 §3.4 source ↗

Write f(x)= 4 3− 4+ x 2 as the composition of two functions.

Answer

Possible answer:

g( x )= 4+ x 2 h( x )= 4 3−x f=h∘g

3.4-1 §3.4 source ↗

How does one find the domain of the quotient of two functions, f g ?

Answer

Find the numbers that make the function in the denominator g equal to zero, and check for any other domain restrictions on f and g, such as an even-indexed root or zeros in the denominator.

3.4-3 §3.4 source ↗

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

Answer

Yes. Sample answer: Let f(x)= x+ 1and g(x)= x− 1. Then f(g(x))= f(x−1)= (x−1)+ 1= x and g(f(x))= g(x+1)= (x+1)− 1= x. So f∘g=g∘f.

3.4-5 §3.4 source ↗

For the following exercises, determine the domain for each function in interval notation.

Given f(x)= x 2 +2x and  g(x)= 6− x 2 , find f+g,f−g,fg, and f g .

Answer

(f+g)(x)= 2x+ 6, domain: (−∞,∞)

(f−g)(x)= 2 x 2 + 2x− 6, domain: (−∞,∞)

(fg)(x)= − x 4 − 2 x 3 + 6 x 2 + 12x, domain: (−∞,∞)

( f g )(x)= x 2 +2x 6− x 2 , domain: (−∞,− 6 )∪(− 6 , 6 )∪( 6 ,∞)

3.4-7 §3.4 source ↗

For the following exercises, determine the domain for each function in interval notation.

Given f(x)=2 x 2 +4x and g(x)= 1 2x , find f+g,f−g,fg, and f g .

Answer

(f+g)(x)= 4 x 3 +8 x 2 +1 2x , domain: (−∞,0)∪(0,∞)

(f−g)(x)= 4 x 3 +8 x 2 −1 2x , domain: (−∞,0)∪(0,∞)

(fg)(x)= x+ 2, domain: (−∞,0)∪(0,∞)

( f g )(x)= 4 x 3 + 8 x 2 , domain: (−∞,0)∪(0,∞)

3.4-9 §3.4 source ↗

For the following exercises, determine the domain for each function in interval notation.

Given f(x)=3 x 2 and g(x)= x−5 , find f+g,f−g,fg, and f g .

Answer

(f+g)(x)= 3 x 2 + x−5 , domain: [5,∞)

(f−g)(x)= 3 x 2 − x−5 , domain: [5,∞)

(fg)(x)= 3 x 2 x−5 , domain: [5,∞)

( f g )(x)= 3 x 2 x−5 , domain: (5,∞)

3.4-11 §3.4 source ↗

For the following exercises, determine the domain for each function in interval notation.

For the following exercise, find the indicated function given f(x)=2 x 2 +1 and g(x)=3x−5.

ⓐ f(g(2))

ⓑ f(g(x))

ⓒ g(f(x))

ⓓ ( g∘g )( x )

ⓔ ( f∘f )( −2 )

Answer

ⓐ 3=  2(9x2 – 30x + 25) +  1 =  18x2 – 60x +  51

ⓑ f(g(x))= 2 ( 3x−5 ) 2 + 1

ⓒ g(f(x))= 3(2 x 2 +1)-5= 6x2-2

ⓓ(g∘g)(x)= 3(3x−5)− 5= 9x− 20

ⓔ (f∘f)(−2)= 163

3.4-13 §3.4 source ↗

For the following exercises, use each pair of functions to find f( g( x ) ) and g( f( x ) ). Simplify your answers.

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

Answer

f(g(x))= x 2 +3 + 2, g(f(x))= x+ 4 x + 7

3.4-15 §3.4 source ↗

For the following exercises, use each pair of functions to find f( g( x ) ) and g( f( x ) ). Simplify your answers.

f(x)= x 3 , g(x)= x+1 x 3

Answer

f(g(x))= x+1 x 3 3 = x+1 3 x , g(f(x))= x 3 +1 x

3.4-17 §3.4 source ↗

For the following exercises, use each pair of functions to find f( g( x ) ) and g( f( x ) ). Simplify your answers.

f(x)= 1 x−4 , g(x)= 2 x + 4

Answer

(f∘g)(x)= 1 2 x +4−4 = x 2 , (g∘f)(x)= 2x− 4

3.4-19 §3.4 source ↗

For the following exercises, use each set of functions to find f( g( h(x) ) ). Simplify your answers.

f(x)= x 2 +1, g(x)= 1 x , and h(x)=x+3

Answer

f(g(h(x)))= ( 1 x+3 ) 2 + 1

3.4-21 §3.4 source ↗

For the following exercises, use each set of functions to find f( g( h(x) ) ). Simplify your answers.

Given f(x)= 2−4x and g(x)=− 3 x , find the following:

ⓐ (g∘f)(x)

ⓑ the domain of (g∘f)(x) in interval notation

Answer

ⓐ (g∘f)(x)= − 3 2−4x

ⓑ ( −∞, 1 2 )

3.4-23 §3.4 source ↗

For the following exercises, use each set of functions to find f( g( h(x) ) ). Simplify your answers.

Given functions p(x)= 1 x and m(x)= x 2 −4, state the domain of each of the following functions using interval notation:

ⓐ p(x) m(x)

ⓑ p(m(x))

ⓒ m(p(x))

Answer

ⓐ (0,2)∪(2,∞);

ⓑ (−∞,−2)∪(2,∞);

ⓒ (0,∞)

3.4-25 §3.4 source ↗

For the following exercises, use each set of functions to find f( g( h(x) ) ). Simplify your answers.

For f(x)= 1 x and g(x)= x−1 , write the domain of (f∘g)(x) in interval notation.

Answer

(1,∞)

3.4-27 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= (x−5) 3

Answer

sample: f(x)= x 3 g(x)=x−5

3.4-29 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= 4 (x+2) 2

Answer

sample: f(x)= 4 x g(x)= (x+2) 2

3.4-31 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= 1 2x−3 3

Answer

sample: f(x)= x 3 g(x)= 1 2x−3

3.4-33 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= 3x−2 x+5 4

Answer

sample: f(x)= x 4 g(x)= 3x−2 x+5

3.4-35 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= 2x+6

Answer

sample: f(x)= x g(x)=2x+6

3.4-37 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= x−1 3

Answer

sample: f(x)= x 3 g(x)=(x−1)

3.4-39 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= 1 (x−2) 3

Answer

sample: f(x)= x 3 g(x)= 1 x−2

3.4-41 §3.4 source ↗

For the following exercises, find functions f(x) and g(x) so the given function can be expressed as h(x)= f(g(x)).

h(x)= 2x−1 3x+4

Answer

sample: f(x)= x g(x)= 2x−1 3x+4

3.4-43 §3.4 source ↗

For the following exercises, use the graphs of f, shown in Figure 4, and g, shown in Figure 5, to evaluate the expressions.

Graph of a function. Figure 4 Graph of a function. Figure 5

f( g(1) )

Answer

2

3.4-45 §3.4 source ↗

For the following exercises, use the graphs of f, shown in Figure 4, and g, shown in Figure 5, to evaluate the expressions.

Graph of a function. Figure 4 Graph of a function. Figure 5

g( f(0) )

Answer

5

3.4-47 §3.4 source ↗

For the following exercises, use the graphs of f, shown in Figure 4, and g, shown in Figure 5, to evaluate the expressions.

Graph of a function. Figure 4 Graph of a function. Figure 5

f( f(4) )

Answer

4

3.4-49 §3.4 source ↗

For the following exercises, use the graphs of f, shown in Figure 4, and g, shown in Figure 5, to evaluate the expressions.

Graph of a function. Figure 4 Graph of a function. Figure 5

g( g(0) )

Answer

0

3.4-51 §3.4 source ↗

For the following exercises, use graphs of f(x), shown in Figure 6, g(x), shown in Figure 7, and h(x), shown in Figure 8, to evaluate the expressions.

A graph of the function f(x) showing a parabola opening upwards, with its vertex located at the origin (0, 0). The parabola passes through points such as (-2, 4) and (2, 4). Figure 6 Graph of a parabola. Figure 7 Graph of a square root function. Figure 8

g( f( 2 ) )

Answer

2

3.4-53 §3.4 source ↗

For the following exercises, use graphs of f(x), shown in Figure 6, g(x), shown in Figure 7, and h(x), shown in Figure 8, to evaluate the expressions.

A graph of the function f(x) showing a parabola opening upwards, with its vertex located at the origin (0, 0). The parabola passes through points such as (-2, 4) and (2, 4). Figure 6 Graph of a parabola. Figure 7 Graph of a square root function. Figure 8

f( g( 1 ) )

Answer

1

3.4-55 §3.4 source ↗

For the following exercises, use graphs of f(x), shown in Figure 6, g(x), shown in Figure 7, and h(x), shown in Figure 8, to evaluate the expressions.

A graph of the function f(x) showing a parabola opening upwards, with its vertex located at the origin (0, 0). The parabola passes through points such as (-2, 4) and (2, 4). Figure 6 Graph of a parabola. Figure 7 Graph of a square root function. Figure 8

h( f( 2 ) )

Answer

4

3.4-57 §3.4 source ↗

For the following exercises, use graphs of f(x), shown in Figure 6, g(x), shown in Figure 7, and h(x), shown in Figure 8, to evaluate the expressions.

A graph of the function f(x) showing a parabola opening upwards, with its vertex located at the origin (0, 0). The parabola passes through points such as (-2, 4) and (2, 4). Figure 6 Graph of a parabola. Figure 7 Graph of a square root function. Figure 8

f( g( f( −2 ) ) )

Answer

4

3.4-59 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 3 to evaluate each expression.

x f(x) g(x)
079
165
256
382
441
508
627
713
894
930
Table 3

f( g( 5 ) )

Answer

9

3.4-61 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 3 to evaluate each expression.

x f(x) g(x)
079
165
256
382
441
508
627
713
894
930
Table 3

g( f( 3 ) )

Answer

4

3.4-63 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 3 to evaluate each expression.

x f(x) g(x)
079
165
256
382
441
508
627
713
894
930
Table 3

f( f( 1 ) )

Answer

2

3.4-65 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 3 to evaluate each expression.

x f(x) g(x)
079
165
256
382
441
508
627
713
894
930
Table 3

g( g( 6 ) )

Answer

3

3.4-67 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 4 to evaluate the expressions.

x f(x) g(x)
−3 11 −8
−2 9 −3
−1 7 0
0 5 1
1 3 0
2 1 −3
3 −1 −8
Table 4

(f∘g)(2)

Answer

11

3.4-69 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 4 to evaluate the expressions.

x f(x) g(x)
−3 11 −8
−2 9 −3
−1 7 0
0 5 1
1 3 0
2 1 −3
3 −1 −8
Table 4

(g∘f)(3)

Answer

0

3.4-71 §3.4 source ↗

For the following exercises, use the function values for fand g shown in Table 4 to evaluate the expressions.

x f(x) g(x)
−3 11 −8
−2 9 −3
−1 7 0
0 5 1
1 3 0
2 1 −3
3 −1 −8
Table 4

(f∘f)(3)

Answer

7

3.4-73 §3.4 source ↗

For the following exercises, use each pair of functions to find f( g( 0 ) ) and g( f(0) ).

f(x)= 5x+ 7, g(x)= 4− 2 x 2

Answer

f(g(0))= 27, g(f(0))= − 94

3.4-75 §3.4 source ↗

For the following exercises, use each pair of functions to find f( g( 0 ) ) and g( f(0) ).

f(x)= 1 x+2 , g(x)= 4x+ 3

Answer

f(g(0))= 1 5 , g(f(0))= 5

3.4-77 §3.4 source ↗

For the following exercises, use the functions f(x)=2 x 2 +1 and g(x)=3x+5 to evaluate or find the composite function as indicated.

f( g(x) )

Answer

18 x 2 +60x+51

3.4-79 §3.4 source ↗

For the following exercises, use the functions f(x)=2 x 2 +1 and g(x)=3x+5 to evaluate or find the composite function as indicated.

(g∘g)(x)

Answer

g∘g(x)= 9x+ 20

3.4-81 §3.4 source ↗

For the following exercises, use f(x)= x 3 +1 and g(x)= x−1 3 .

Find (f∘g)(2) and (g∘f)(2).

Answer

2

3.4-83 §3.4 source ↗

For the following exercises, use f(x)= x 3 +1 and g(x)= x−1 3 .

What is the domain of (f∘g)(x)?

Answer

(−∞,∞)

3.4-85 §3.4 source ↗

For the following exercises, let F(x)= (x+1) 5 , f(x)= x 5 , and g(x)=x+1.

True or False: (g∘f)(x)= F(x).

Answer

False

3.4-87 §3.4 source ↗

For the following exercises, find the composition when f(x)= x 2 +2 for all x≥0 and g(x)= x−2 .

(f∘g)(6); (g∘f)(6)

Answer

(f∘g)(6)=6 ; (g∘f)(6)=6

3.4-89 §3.4 source ↗

For the following exercises, find the composition when f(x)= x 2 +2 for all x≥0 and g(x)= x−2 .

(f∘g)(11); (g∘f)(11)

Answer

(f∘g)(11)= 11, (g∘f)(11)= 11

3.4-91 §3.4 source ↗

For the following exercises, find the composition when f(x)= x 2 +2 for all x≥0 and g(x)= x−2 .

The function A(d) gives the pain level on a scale of 0 to 10 experienced by a patient with d milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after t minutes is modeled by m(t). Which of the following would you do in order to determine when the patient will be at a pain level of 4?

ⓐEvaluate A( m(4) ).

ⓑEvaluate m( A(4) ).

ⓒSolve A( m(t) )=4.

ⓓSolve m( A(d) )=4.

Answer

c

3.4-93 §3.4 source ↗

For the following exercises, find the composition when f(x)= x 2 +2 for all x≥0 and g(x)= x−2 .

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to r(t)=25 t+2 , find the area of the ripple as a function of time. Find the area of the ripple at t=2.

Answer

A(t)= π ( 25 t+2 ) 2 and A(2)= π ( 25 4 ) 2 = 2500π square inches

3.4-95 §3.4 source ↗

For the following exercises, find the composition when f(x)= x 2 +2 for all x≥0 and g(x)= x−2 .

Use the function you found in the previous exercise to find the total area burned after 5 minutes.

Answer

A(5)= π ( 2(5)+1 ) 2 = 121π square units

3.4-97 §3.4 source ↗

For the following exercises, find the composition when f(x)= x 2 +2 for all x≥0 and g(x)= x−2 .

The number of bacteria in a refrigerated food product is given by N(T)= 23 T 2 − 56T+ 1, 3< T< 33, where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t)= 5t+ 1.5, where t is the time in hours.

ⓐFind the composite function N( T(t) ).

ⓑFind the time (round to two decimal places) when the bacteria count reaches 6752.

Answer

ⓐ N(T(t))= 23 (5t+1.5) 2 − 56(5t+1.5)+ 1

ⓑ 3.38 hours

Try It #1 §3.5 source ↗

The function h(t)= − 4.9 t 2 + 30t gives the height h of a ball (in meters) thrown upward from the ground after t seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function b(t) to h(t), and then find a formula for b(t).

Answer

b(t)=h(t)+10=−4.9 t 2 +30t+10

Try It #2 §3.5 source ↗

Given the function f(x)= x , graph the original function f(x) and the transformation g(x)=f(x+2) on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

Answer

The graphs of f(x) and g(x) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

Graph of a square root function and a horizontally shift square foot function.

Try It #3 §3.5 source ↗

Given f(x)=| x |, sketch a graph of h(x)= f(x−2)+ 4.

Answer

Graph of h(x)=|x-2|+4.

Try It #4 §3.5 source ↗

Write a formula for a transformation of the toolkit reciprocal function f( x )= 1 x that shifts the function’s graph one unit to the right and one unit up.

Answer

g( x )= 1 x-1 +1

Try It #5 §3.5 source ↗

Reflect the graph of f(x)=|x−1| (a) vertically and (b) horizontally.

Answer

ⓐ Graph of a vertically reflected absolute function.

ⓑ Graph of an absolute function translated one unit left.

Try It #6 §3.5 source ↗

A function f(x) is given as Table 9. Create a table for the functions below.

ⓐ g(x)=−f(x)

ⓑ h(x)=f(−x)

x −2 0 2 4
f(x) 5 10 15 20
Table 9

Answer

ⓐ

g(x)=−f(x)

x -2 0 2 4
g(x) −5 −10 −15 −20

ⓑ

h(x)=f(−x)

x -2 0 2 4
h(x) 15 10 5 unknown

Try It #7 §3.5 source ↗

Given the toolkit function f(x)= x 2 , graph g(x)=−f(x) and h(x)=f(−x). Take note of any surprising behavior for these functions.

Answer

Graph of x^2 and its reflections.

Notice: g(x)=f(−x) looks the same as f(x) .

Try It #8 §3.5 source ↗

Is the function f(s)= s 4 + 3 s 2 + 7 even, odd, or neither?

Answer

even

Try It #9 §3.5 source ↗

A function f is given as Table 12. Create a table for the function g(x)= 3 4 f(x).

x 2 4 6 8
f(x) 12 16 20 0
Table 12

Answer

x 2 4 6 8
g(x) 9 12 15 0

Try It #10 §3.5 source ↗

Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.

Answer

g(x)=3x-2

Try It #11 §3.5 source ↗

Write a formula for the toolkit square root function horizontally stretched by a factor of 3.

Answer

g(x)=f( 1 3 x ) so using the square root function we get g(x)= 1 3 x

3.5-1 §3.5 source ↗

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

Answer

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

3.5-3 §3.5 source ↗

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

Answer

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

3.5-5 §3.5 source ↗

How can you determine whether a function is odd or even from the formula of the function?

Answer

For a function f, substitute (−x) for (x) in f(x). Simplify. If the resulting function is the same as the original function, f(−x)=f(x), then the function is even. If the resulting function is the opposite of the original function, f(−x)= − f(x), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

3.5-7 §3.5 source ↗

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

f(x)=| x | is shifted down 3 units and to the right 1 unit.

Answer

g(x)= |x-1|− 3

3.5-9 §3.5 source ↗

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

f(x)= 1 x 2 is shifted up 2 units and to the left 4 units.

Answer

g(x)= 1 (x+4) 2 + 2

3.5-11 §3.5 source ↗

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f.

y=f(x+43)

Answer

The graph of f(x+43) is a horizontal shift to the left 43 units of the graph of f.

3.5-13 §3.5 source ↗

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f.

y=f(x−4)

Answer

The graph of f(x-4) is a horizontal shift to the right 4 units of the graph of f.

3.5-15 §3.5 source ↗

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f.

y=f(x)+8

Answer

The graph of f(x)+8 is a vertical shift up 8 units of the graph of f.

3.5-17 §3.5 source ↗

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f.

y=f(x)−7

Answer

The graph of f(x)−7 is a vertical shift down 7 units of the graph of f.

3.5-19 §3.5 source ↗

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f.

y=f(x+4)−1

Answer

The graph of f(x+4)−1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f.

3.5-21 §3.5 source ↗

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

g(x)= 5 (x+3) 2 − 2

Answer

decreasing on (−∞,−3) and increasing on (−3,∞)

3.5-23 §3.5 source ↗

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

k(x)=−3 x −1

Answer

decreasing on (0,∞)

3.5-25 §3.5 source ↗

For the following exercises, use the graph of f(x)= 2 x shown in Figure 31 to sketch a graph of each transformation of f(x).

Graph of f(x). Figure 31

h(x)= 2 x −3

Answer

A graph displays an exponential function h, characterized by a left-to-right increasing curve. It approaches a horizontal asymptote at y=-3 for negative x, and rises steeply for positive x.

3.5-27 §3.5 source ↗

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

f(t)= (t+1) 2 − 3

Answer

Graph of f(t).

3.5-29 §3.5 source ↗

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

k(x)= (x−2) 3 − 1

Answer

Graph of k(x).

3.5-31 §3.5 source ↗

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

Tabular representations for the functions f,g, and h are given below. Write g(x) and h(x) as transformations of f(x).

x −2 −1 0 1 2
f(x) −2 −1 −3 1 2
x −1 0 1 2 3
g(x) −2 −1 −3 1 2
x −2 −1 0 1 2
h(x) −1 0 −2 2 3

Answer

g(x)= f(x-1), h(x)= f(x)+ 1

3.5-33 §3.5 source ↗

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

Graph of an absolute function.

Answer

f(x)= |x-3|− 2

3.5-35 §3.5 source ↗

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

Graph of a square root function.

Answer

f(x)= x+3 −1

3.5-37 §3.5 source ↗

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

Graph of a parabola

Answer

f(x)= (x-2) 2

3.5-39 §3.5 source ↗

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

Graph of an absolute function.

Answer

f(x)= |x+ 3|− 2

3.5-41 §3.5 source ↗

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

Graph of a square root function.

Answer

f(x)=− x

3.5-43 §3.5 source ↗

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

Graph of a parabola.

Answer

f(x)= − (x+1) 2 + 2

3.5-45 §3.5 source ↗

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

Graph of a square root function.

Answer

f(x)= −x +1

3.5-47 §3.5 source ↗

For the following exercises, determine whether the function is odd, even, or neither.

f(x)=3 x 4

Answer

even

3.5-49 §3.5 source ↗

For the following exercises, determine whether the function is odd, even, or neither.

h(x)= 1 x +3x

Answer

odd

3.5-51 §3.5 source ↗

For the following exercises, determine whether the function is odd, even, or neither.

g(x)=2 x 4

Answer

even

3.5-53 §3.5 source ↗

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f.

g(x)=−f(x)

Answer

The graph of g is a vertical reflection (across the x -axis) of the graph of f.

3.5-55 §3.5 source ↗

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f.

g(x)=4f(x)

Answer

The graph of g is a vertical stretch by a factor of 4 of the graph of f.

3.5-57 §3.5 source ↗

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f.

g(x)=f(5x)

Answer

The graph of g is a horizontal compression by a factor of 1 5 of the graph of f.

3.5-59 §3.5 source ↗

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f.

g(x)=f( 1 3 x )

Answer

The graph of g is a horizontal stretch by a factor of 3 of the graph of f.

3.5-61 §3.5 source ↗

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f.

g(x)=3f( −x )

Answer

The graph of g is a horizontal reflection across the y -axis and a vertical stretch by a factor of 3 of the graph of f.

3.5-63 §3.5 source ↗

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described.

The graph of f(x)=|x| is reflected over the y -axis and horizontally compressed by a factor of 1 4 .

Answer

g(x)=|−4x|

3.5-65 §3.5 source ↗

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described.

The graph of f(x)= 1 x 2 is vertically compressed by a factor of 1 3 , then shifted to the left 2 units and down 3 units.

Answer

g(x)= 1 3 (x+2) 2 − 3

3.5-67 §3.5 source ↗

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described.

The graph of f(x)= x 2 is vertically compressed by a factor of 1 2 , then shifted to the right 5 units and up 1 unit.

Answer

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

3.5-69 §3.5 source ↗

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

g(x)= 4 (x+1) 2 − 5

Answer

The graph of the function f(x)= x 2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

Graph of a parabola.

3.5-71 §3.5 source ↗

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

h(x)= − 2|x− 4|+ 3

Answer

The graph of f(x)=|x| is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

Graph of an absolute function.

3.5-73 §3.5 source ↗

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

m(x)= 1 2 x 3

Answer

The graph of the function f(x)= x 3 is compressed vertically by a factor of 1 2 .

Graph of a cubic function.

3.5-75 §3.5 source ↗

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

p(x)= ( 1 3 x ) 3 − 3

Answer

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

Graph of a cubic function.

3.5-77 §3.5 source ↗

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

a(x)= −x+4

Answer

The graph of f(x)= x is reflected across the y-axis and then shifted right 4 units.

Graph of a square root function.

3.5-79 §3.5 source ↗

For the following exercises, use the graph in Figure 32 to sketch the given transformations.

Graph of a polynomial. Figure 32

g(x)=−f(x)

Answer

Graph of a polynomial.

3.5-81 §3.5 source ↗

For the following exercises, use the graph in Figure 32 to sketch the given transformations.

Graph of a polynomial. Figure 32

g(x)=f(x−2)

Answer

Graph of a polynomial.

Try It #1 §3.6 source ↗

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Answer

using the variable p for passing, | p−80 |≤20

Try It #2 §3.6 source ↗

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.

Answer

f(x)= − |x+ 2|+ 3

Try It #3 §3.6 source ↗

For the function f(x)= |2x− 1|− 3, find the values of x such that f(x)=0.

Answer

x=−1 or x=2

3.6-1 §3.6 source ↗

How do you solve an absolute value equation?

Answer

Isolate the absolute value term so that the equation is of the form |A|=B. Form one equation by setting the expression inside the absolute value symbol, A, equal to the expression on the other side of the equation, B. Form a second equation by setting A equal to the opposite of the expression on the other side of the equation, −B. Solve each equation for the variable.

3.6-3 §3.6 source ↗

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

Answer

The graph of the absolute value function does not cross the x -axis, so the graph is either completely above or completely below the x -axis.

3.6-5 §3.6 source ↗

Describe all numbers x that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

Answer

The distance from x to 8 can be represented using the absolute value statement: ∣ x − 8 ∣ = 4.

3.6-7 §3.6 source ↗

Describe the situation in which the distance that point x is from 10 is at least 15 units. Express this set of numbers using absolute value notation.

Answer

∣ x − 10 ∣ ≥ 15

3.6-9 §3.6 source ↗

For the following exercises, find the x- and y-intercepts of the graphs of each function.

f(x)= 4|x− 3|+ 4

Answer

There are no x-intercepts.

3.6-11 §3.6 source ↗

For the following exercises, find the x- and y-intercepts of the graphs of each function.

f(x)= − 2|x+ 1|+ 6

Answer

(−4, 0) and (2, 0)

3.6-13 §3.6 source ↗

For the following exercises, find the x- and y-intercepts of the graphs of each function.

f(x)= 2|x− 1|− 6

Answer

(0,−4), (4,0), (−2,0)

3.6-15 §3.6 source ↗

For the following exercises, find the x- and y-intercepts of the graphs of each function.

f(x)= − |x− 9|+ 16

Answer

(0,7), (25,0), (−7,0)

3.6-17 §3.6 source ↗

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

y=|x+1|

Answer

A V-shaped graph with a vertex at (-1, 0), showing points (-3, 2), (-2, 1), (0, 1), and (1, 2).

3.6-19 §3.6 source ↗

For the following exercises, graph the given functions by hand.

y=| x |−2

Answer

A V-shaped graph is displayed on a Cartesian coordinate system. The x-axis ranges from -5 to 4, and the y-axis ranges from -3 to 5. The vertex of the graph is at the point (0, -2). The graph opens upwards, passing through the x-axis at (-2, 0) and (2, 0). The line segment for x less than 0 has a slope of -1, and the line segment for x greater than 0 has a slope of 1. Arrows on both ends indicate that the graph extends indefinitely, representing the function y = |x| - 2.

3.6-21 §3.6 source ↗

For the following exercises, graph the given functions by hand.

y=−| x |−2

Answer

A graph showing a V-shaped function on a Cartesian coordinate system, with its vertex located at (0, -2).

3.6-23 §3.6 source ↗

For the following exercises, graph the given functions by hand.

f(x)= − |x− 1|− 2

Answer

A graph on a coordinate plane displays a V-shaped function. The vertex of the V is located at the point (0, -2). From this vertex, one arm of the V extends downwards and to the left, passing through approximately (-2, -4), and the other arm extends downwards and to the right, passing through approximately (2, -4). Both arms are indicated with arrows, signifying that the function continues indefinitely in those directions. The coordinate plane shows x-axis values from -5 to 5 and y-axis values from -7 to 5, with grid lines at each integer increment.

3.6-25 §3.6 source ↗

For the following exercises, graph the given functions by hand.

f(x)= 2|x+ 3|+ 1

Answer

A graph showing an absolute value function in a V-shape. The vertex of the function is located at the point (-3, 1), and the graph opens upwards on a coordinate plane.

3.6-27 §3.6 source ↗

For the following exercises, graph the given functions by hand.

f(x)= |2x− 4|− 3

Answer

A graph displays an absolute value function, forming a V-shape on a Cartesian coordinate plane. The x-axis is labeled from -6 to 5, and the y-axis is labeled from -6 to 6. The graph has its vertex at (2, -3) and opens upwards, passing through points such as (0, 1), (1, -1), (3, -1), and (4, 1).

3.6-29 §3.6 source ↗

For the following exercises, graph the given functions by hand.

f(x)= − |x− 1|− 3

Answer

A coordinate plane shows an absolute value function graphed as a V-shape opening downwards. The x-axis ranges from -6 to 5, and the y-axis ranges from -6 to 6. The vertex of the V-shape is located at the point (1, -3). The two linear segments extend downwards and outwards from the vertex, with arrows indicating that the graph continues indefinitely.

3.6-31 §3.6 source ↗

For the following exercises, graph the given functions by hand.

f(x)= 1 2 |x+ 4|− 3

Answer

A graph on a coordinate plane shows a V-shaped function. The vertex of the V is located at the point (-4, -3). From this vertex, two arms extend upwards, with arrows indicating they continue infinitely.

3.6-33 §3.6 source ↗

For the following exercises, graph the given functions by hand.

Use a graphing utility to graph f(x)= − 100|x|+ 100 on the viewing window [ −5,5 ]. Identify the corresponding range. Show the graph.

Answer

range: [–400,100]

Graph of an absolute function.

3.6-35 §3.6 source ↗

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

f(x)= 4× 10 9 |x− (5× 10 9 )|+ 2× 10 9

Answer

Graph of an absolute function.

3.6-37 §3.6 source ↗

For the following exercises, solve the inequality.

If possible, find all values of a such that there are no y -intercepts for f(x)= 2|x+ 1|+ a.

Answer

There is no solution for a that will keep the function from having a y -intercept. The absolute value function always crosses the y -intercept when x=0.

3.6-39 §3.6 source ↗

For the following exercises, solve the inequality.

The true proportion p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Answer

|p− 0.08|≤0.015

3.6-41 §3.6 source ↗

For the following exercises, solve the inequality.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x as the diameter of the bearing, write this statement using absolute value notation.

Answer

|x− 5.0|≤0.01

Try It #1 §3.7 source ↗

Given that h −1 (6)=2, what are the corresponding input and output values of the original function h?

Answer

h(2)=6

Try It #2 §3.7 source ↗

If f( x )= x 3 −4 and g( x )= x+4 3 , is g= f −1 ?

Answer

Yes

Try It #3 §3.7 source ↗

If f(x)= ( x−1 ) 3 andg(x)= x 3 + 1, is g= f −1 ?

Answer

Yes

Try It #4 §3.7 source ↗

The domain of function f is (1,∞) and the range of function f is (−∞,−2). Find the domain and range of the inverse function.

Answer

The domain of function f −1 is (−∞,−2) and the range of function f −1 is (1,∞).

Try It #5 §3.7 source ↗

Using Table 4, find and interpret ⓐ f(60), and ⓑ f −1 (60).

t(minutes) 30 50 60 70 90
f( t )(miles) 20 40 50 60 70
Table 4

Answer

ⓐ f(60)=50. In 60 minutes, 50 miles are traveled.

ⓑ f −1 (60)=70. To travel 60 miles, it will take 70 minutes.

Try It #6 §3.7 source ↗

Using the graph in Figure 5, ⓐ find g −1 (1), and ⓑ estimate g −1 (4).

Answer

ⓐ 3

ⓑ 5.6

Try It #7 §3.7 source ↗

Solve for x in terms of y given y= 1 3 (x−5).

Answer

x=3y+5

Try It #8 §3.7 source ↗

What is the inverse of the function f(x)=2− x ? State the domains of both the function and the inverse function.

Answer

f −1 (x)= ( 2−x ) 2 ; domainoff:[ 0,∞ ); domainof f −1 :( −∞,2 ]

Try It #9 §3.7 source ↗

Draw graphs of the functions f and f −1 from Example 8.

Answer

Graph of f(x) and f^(-1)(x).

3.7-1 §3.7 source ↗

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Answer

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y -values repeat and the function is one-to-one.

3.7-3 §3.7 source ↗

Can a function be its own inverse? Explain.

Answer

Yes. For example, f(x)= 1 x is its own inverse.

3.7-5 §3.7 source ↗

How do you find the inverse of a function algebraically?

Answer

Given a function y=f(x), solve for x in terms of y. Interchange the x and y. Solve the new equation for y. The expression for y is the inverse, y= f −1 (x).

3.7-7 §3.7 source ↗

For the following exercises, find f −1 (x) for each function.

f(x)=x+3

Answer

f −1 (x)=x−3

3.7-9 §3.7 source ↗

For the following exercises, find f −1 (x) for each function.

f(x)=2−x

Answer

f −1 (x)=2−x

3.7-11 §3.7 source ↗

For the following exercises, find f −1 (x) for each function.

f(x)= x x+2

Answer

f −1 (x)= −2x x−1

3.7-13 §3.7 source ↗

For the following exercises, find a domain on which each function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

f(x)= (x+7) 2

Answer

domain of f(x):[−7,∞); f −1 (x)= x − 7

3.7-15 §3.7 source ↗

For the following exercises, find a domain on which each function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

f(x)= x 2 −5

Answer

domain of f(x):[0,∞); f −1 (x)= x+5

3.7-16 §3.7 source ↗

For the following exercises, find a domain on which each function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

Given f( x )= x 2+x and g(x)= 2x 1−x :

ⓐFind f(g(x)) and g(f(x)).

ⓑWhat does the answer tell us about the relationship between f(x) and g(x)?

Answer

ⓐ f(g(x))=x and g(f(x))=x.

ⓑThis tells us that f and g are inverse functions

3.7-17 §3.7 source ↗

For the following exercises, use function composition to verify that f(x) and g(x) are inverse functions.

f(x)= x−1 3 and g(x)= x 3 +1

Answer

 f(g(x))= x, g(f(x))= x

3.7-19 §3.7 source ↗

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f(x)= x

Answer

one-to-one

3.7-21 §3.7 source ↗

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f(x)=−5x+1

Answer

one-to-one

3.7-23 §3.7 source ↗

For the following exercises, determine whether the graph represents a one-to-one function.

Graph of a parabola.

Answer

not one-to-one

3.7-25 §3.7 source ↗

For the following exercises, use the graph of f shown in Figure 11.

Graph of a line. Figure 11

Find f( 0 ).

Answer

3

3.7-27 §3.7 source ↗

For the following exercises, use the graph of f shown in Figure 11.

Graph of a line. Figure 11

Find f −1 ( 0 ).

Answer

2

3.7-29 §3.7 source ↗

For the following exercises, use the graph of the one-to-one function shown in Figure 12.

Graph of a square root function. Figure 12

Sketch the graph of f −1 .

Answer

Graph of a square root function and its inverse.

3.7-31 §3.7 source ↗

For the following exercises, use the graph of the one-to-one function shown in Figure 12.

Graph of a square root function. Figure 12

If the complete graph of f is shown, find the domain of f.

Answer

[ 2,10 ]

3.7-33 §3.7 source ↗

For the following exercises, evaluate or solve, assuming that the function f is one-to-one.

If f(6)=7, find f −1 (7).

Answer

6

3.7-35 §3.7 source ↗

For the following exercises, evaluate or solve, assuming that the function f is one-to-one.

If f −1 ( −4 )=−8, find f(−8).

Answer

−4

3.7-37 §3.7 source ↗

For the following exercises, use the values listed in Table 6 to evaluate or solve.

x 0 1 2 3 4 5 6 7 8 9
f(x) 8 0 7 4 2 6 5 3 9 1
Table 6

Find f( 1 ).

Answer

0

3.7-39 §3.7 source ↗

For the following exercises, use the values listed in Table 6 to evaluate or solve.

x 0 1 2 3 4 5 6 7 8 9
f(x) 8 0 7 4 2 6 5 3 9 1
Table 6

Find f −1 ( 0 ).

Answer

1

3.7-41 §3.7 source ↗

For the following exercises, use the values listed in Table 6 to evaluate or solve.

x 0 1 2 3 4 5 6 7 8 9
f(x) 8 0 7 4 2 6 5 3 9 1
Table 6

Use the tabular representation of f in Table 7 to create a table for f −1 ( x ).

x 3 6 9 13 14
f(x) 1 4 7 12 16
Table 7

Answer

x 1 4 7 12 16
f −1 (x) 3 6 9 13 14

3.7-43 §3.7 source ↗

For the following exercises, find the inverse function. Then, graph the function and its inverse.

f(x)= x 3 −1

Answer

f −1 (x)= (1+x) 1/3

Graph of a cubic function and its inverse.

3.7-45 §3.7 source ↗

For the following exercises, find the inverse function. Then, graph the function and its inverse.

To convert from x degrees Celsius to y degrees Fahrenheit, we use the formula f(x)= 9 5 x+ 32. Find the inverse function, if it exists, and explain its meaning.

Answer

f −1 (x)= 5 9 (x−32). Given the Fahrenheit temperature, x, this formula allows you to calculate the Celsius temperature.

3.7-47 §3.7 source ↗

For the following exercises, find the inverse function. Then, graph the function and its inverse.

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, t, in hours given by d(t)=50t. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function t(d). Find t(180) and interpret its meaning.

Answer

t(d)= d 50 , t(180)= 180 50 . The time for the car to travel 180 miles is 3.6 hours.

3.R-1 §3.R source ↗

For the following exercises, determine whether the relation is a function.

{ (a,b),(c,d),(e,d) }

Answer

function

3.R-3 §3.R source ↗

For the following exercises, determine whether the relation is a function.

y 2 +4=x, for x the independent variable and y the dependent variable

Answer

not a function

3.R-5 §3.R source ↗

For the following exercises, evaluate f(−3); f(2); f(−a); − f(a); f(a+h).

f(x)= − 2 x 2 + 3x

Answer

f(−3)=−27; f(2)=−2; f(−a)= − 2 a 2 − 3a;
− f(a)= 2 a 2 − 3a; f(a+h)= − 2 a 2 + 3a− 4ah+ 3h− 2 h 2

3.R-7 §3.R source ↗

For the following exercises, determine whether the functions are one-to-one.

f(x)=−3x+5

Answer

one-to-one

3.R-9 §3.R source ↗

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

Graph of a cubic function.

Answer

function

3.R-11 §3.R source ↗

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

Graph of a relation.

Answer

function

3.R-13 §3.R source ↗

For the following exercises, graph the functions.

f(x)= x 2 −2

Answer

A graph of an upward-opening parabola with its vertex at (0, -2) and passing through approximately (-1.4, 0) and (1.4, 0). The curve extends upwards on both sides.

3.R-15 §3.R source ↗

For the following exercises, use Figure 2 to approximate the values.

Graph of a parabola. Figure 2

f(−2)

Answer

2

3.R-17 §3.R source ↗

For the following exercises, use Figure 2 to approximate the values.

Graph of a parabola. Figure 2

If f(x)=1, then solve for x.

Answer

x=−1.8 or or x= 1.8

3.R-19 §3.R source ↗

For the following exercises, use the function h(t)= − 16 t 2 + 80t to find the values in simplest form.

h(a)−h(1) a−1

Answer

−64+80a−16 a 2 −1+a = − 16a+ 64

3.R-21 §3.R source ↗

For the following exercises, find the domain of each function, expressing answers using interval notation.

f(x)= x−3 x 2 −4x−12

Answer

( −∞,−2 )∪( −2,6 )∪( 6,∞ )

3.R-23 §3.R source ↗

For the following exercises, find the domain of each function, expressing answers using interval notation.

Graph this piecewise function: f(x)={ x+1       x<−2 −2x−3  x≥−2

Answer

A piecewise linear graph featuring a jump discontinuity at x = -2. The function approaches -1 from the left with an open circle, and has a value of 1 from the right, continuing downwards.

3.R-25 §3.R source ↗

For the following exercises, find the average rate of change of the functions from x= 1to x= 2.

f(x)=10 x 2 +x

Answer

31

3.R-27 §3.R source ↗

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Graph of a parabola.

Answer

increasing ( 2,∞ ); decreasing (−∞,2)

3.R-29 §3.R source ↗

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Graph of a function.

Answer

increasing ( −3,1 ); constant (−∞,−3)∪( 1,∞ )

3.R-31 §3.R source ↗

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Find the local extrema for the function graphed in Exercise 3.28.

Answer

local minimum ( −2,−3 ); local maximum ( 1,3 )

3.R-33 §3.R source ↗

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Find the absolute maximum of the function graphed in Figure 3.

Graph of a cubic function. Figure 3

Answer

( −1.8,10 )

3.R-35 §3.R source ↗

For the following exercises, find (f∘g)(x) and (g∘f)(x) for each pair of functions.

f(x)= 3x+ 2, g(x)= 5− 6x

Answer

(f∘g)(x)= 17− 18x; (g∘f)(x)= − 7− 18x

3.R-37 §3.R source ↗

For the following exercises, find (f∘g)(x) and (g∘f)(x) for each pair of functions.

f(x)= x+2 , g(x)= 1 x

Answer

(f∘g)(x)= 1 x +2 ; (g∘f)(x)= 1 x+2

3.R-39 §3.R source ↗

For the following exercises, find ( f∘g ) and the domain for ( f∘g )(x) for each pair of functions.

f(x)= x+1 x+4 , g(x)= 1 x

Answer

(f∘g)(x)= 1+x 1+4x ,  x≠0,  x≠− 1 4

3.R-41 §3.R source ↗

For the following exercises, find ( f∘g ) and the domain for ( f∘g )(x) for each pair of functions.

f(x)= 1 x , g(x)= x

Answer

(f∘g)(x)= 1 x , x> 0

3.R-43 §3.R source ↗

For the following exercises, express each function H as a composition of two functions f and g where H(x)= (f∘g)(x).

H(x)= 2x−1 3x+4

Answer

sample: g(x)= 2x−1 3x+4 ; f(x)= x

3.R-45 §3.R source ↗

For the following exercises, sketch a graph of the given function.

f(x)= (x−3) 2

Answer

Graph of f(x)

3.R-47 §3.R source ↗

For the following exercises, sketch a graph of the given function.

f(x)= x +5

Answer

Graph of f(x)

3.R-49 §3.R source ↗

For the following exercises, sketch a graph of the given function.

f(x)= −x 3

Answer

Graph of f(x)

3.R-51 §3.R source ↗

For the following exercises, sketch a graph of the given function.

f(x)= 4[|x−2|−6]

Answer

A V-shaped graph resembling an absolute value function is shown on a Cartesian plane. Its vertex is at (2, -24), and it crosses the x-axis at (-4, 0) and (8, 0).

3.R-53 §3.R source ↗

For the following exercises, sketch the graph of the function g if the graph of the function f is shown in Figure 4.

This graph shows a semi-circle centered at the origin with a radius of 2. The curve originates at (-2, 0), reaches its apex at (0, 2), and terminates at (2, 0) on the Cartesian plane. Figure 4

g(x)=f(x−1)

Answer

Graph of a half circle.

3.R-55 §3.R source ↗

For the following exercises, write the equation for the standard function represented by each of the graphs below.

Graph of an absolute function.

Answer

f(x)=| x−3 |

3.R-57 §3.R source ↗

For the following exercises, determine whether each function below is even, odd, or neither.

f(x)=3 x 4

Answer

even

3.R-59 §3.R source ↗

For the following exercises, determine whether each function below is even, odd, or neither.

h(x)= 1 x +3x

Answer

odd

3.R-61 §3.R source ↗

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

Graph of a parabola.

Answer

even

3.R-63 §3.R source ↗

For the following exercises, write an equation for the transformation of f(x)=| x |.

Graph of f(x).

Answer

f(x)= 1 2 |x+ 2|+ 1

3.R-65 §3.R source ↗

For the following exercises, write an equation for the transformation of f(x)=| x |.

Graph of f(x).

Answer

f(x)= − 3|x− 3|+ 3

3.R-67 §3.R source ↗

For the following exercises, graph the absolute value function.

f(x)=−| x−3 |

Answer

A graph displays a blue V-shaped function, symmetrical around x=3, with its vertex at (3, 0). The two linear segments extend downwards, passing through points (0, -3) and (6, -3).

3.R-69 §3.R source ↗

For the following exercises, find f −1 (x) for each function.

f(x)=9+10x

Answer

f −1 (x)= x-9 10

3.R-71 §3.R source ↗

For the following exercise, find a domain on which the function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

f(x)= x 2 +1

Answer

f −1 (x) = x-1

3.R-73 §3.R source ↗

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f(x)= 1 x

Answer

The function is one-to-one.

A graph of the reciprocal function y = 1/x, showing a curve in the first and third quadrants of the Cartesian coordinate system. The x-axis and y-axis are labeled with tick marks from -4 to 4. The curve has a vertical asymptote at x=0 (the y-axis) and a horizontal asymptote at y=0 (the x-axis). In the first quadrant, the curve starts high near the positive y-axis, passes through (1,1) and extends towards the positive x-axis. In the third quadrant, the curve starts low near the negative y-axis, passes through (-1,-1) and extends towards the negative x-axis.

3.R-75 §3.R source ↗

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

If f( 5 )=2, find f −1 (2).

Answer

5

3.T-1 §3.T source ↗

For the following exercises, determine whether each of the following relations is a function.

y=2x+8

Answer

The relation is a function.

3.T-3 §3.T source ↗

For the following exercises, evaluate the function f(x)= − 3 x 2 + 2x at the given input.

f(−2)

Answer

−16

3.T-5 §3.T source ↗

For the following exercises, evaluate the function f(x)= − 3 x 2 + 2x at the given input.

Show that the function f(x)= − 2 (x−1) 2 + 3 is not one-to-one.

Answer

The graph is a parabola and the graph fails the horizontal line test.

3.T-7 §3.T source ↗

For the following exercises, evaluate the function f(x)= − 3 x 2 + 2x at the given input.

Given f(x)= 2 x 2 − 5x, find f(a+1)−f(1) in simplest form.

Answer

2 a 2 −a

3.T-9 §3.T source ↗

For the following exercises, evaluate the function f(x)= − 3 x 2 + 2x at the given input.

Find the average rate of change of the function f(x)= 3− 2 x 2 + x by finding f(b)−f(a) b−a in simplest form.

Answer

−2(a+b)+1

3.T-11 §3.T source ↗

For the following exercises, use the functions f(x)= 3− 2 x 2 + xand g(x)= x to find the composite functions.

( g∘f )(1)

Answer

2

3.T-13 §3.T source ↗

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

f(x)= x+6 −1

Answer

A graph displays a curve resembling a square root function, starting at approximately (-6, -1) and extending upward and to the right, passing through (-5, 0).

3.T-15 §3.T source ↗

For the following exercises, determine whether the functions are even, odd, or neither.

f(x)= − 5 x 2 + 9 x 6

Answer

even

3.T-17 §3.T source ↗

For the following exercises, determine whether the functions are even, odd, or neither.

f(x)= 1 x

Answer

odd

3.T-19 §3.T source ↗

For the following exercises, find the inverse of the function.

f(x)=3x−5

Answer

f −1 (x)= x+5 3

3.T-21 §3.T source ↗

For the following exercises, use the graph of g shown in Figure 1.

Graph of a cubic function. Figure 1

On what intervals is the function increasing?

Answer

(−∞,−1.1)and (1.1,∞)

3.T-23 §3.T source ↗

For the following exercises, use the graph of g shown in Figure 1.

Graph of a cubic function. Figure 1

Approximate the local minimum of the function. Express the answer as an ordered pair. The ordered pair should include both the x value as well as the g(x) value.

Answer

( 1.1,−0.9 )

3.T-25 §3.T source ↗

For the following exercises, use the graph of the piecewise function shown in Figure 2.

Graph of absolute function and step function. Figure 2

Find f(2).

Answer

f(2)=2

3.T-27 §3.T source ↗

For the following exercises, use the graph of the piecewise function shown in Figure 2.

Graph of absolute function and step function. Figure 2

Write an equation for the piecewise function.

Answer

f(x)={ | x |ifx≤2 3ifx>2

3.T-29 §3.T source ↗

For the following exercises, use the values listed in Table 1.

x 0 1 2 3 4 5 6 7 8
F(x) 1 3 5 7 9 11 13 15 17
Table 1

Solve the equation f(x)=5.

Answer

x=2

3.T-31 §3.T source ↗

For the following exercises, use the values listed in Table 1.

x 0 1 2 3 4 5 6 7 8
F(x) 1 3 5 7 9 11 13 15 17
Table 1

Is the function represented by the graph one-to-one?

Answer

yes

3.T-33 §3.T source ↗

For the following exercises, use the values listed in Table 1.

x 0 1 2 3 4 5 6 7 8
F(x) 1 3 5 7 9 11 13 15 17
Table 1

Given f(x)= − 2x+ 11, find f −1 (x).

Answer

f −1 (x)= − x−11 2

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Problems and answers are adapted from Algebra and Trigonometry 2e by Jay Abramson, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.