Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 10

Nonlinear Functions

Educators


Problem 1

Which of the following rules define $y$ as a function of $x ?$

Linh V.
Numerade Educator

Problem 2

Which of the following rules define $y$ as a function of $x ?$

Linh V.
Numerade Educator

Problem 3

Nonlinear Functions
$$\begin{array}{|c|c|}\hline x & {y} \\ \hline 3 & {9} \\ \hline 2 & {4} \\ \hline 1 & {1} \\ \hline 0 & {0} \\ \hline-1 & {1} \\ \hline-2 & {4} \\ \hline-3 & {9} \\ \hline\end{array}$$

Linh V.
Numerade Educator

Problem 4

Nonlinear Functions
$$
\begin{array}{|r|r|}\hline x & {y} \\ \hline 9 & {3} \\ \hline 4 & {2} \\ \hline 1 & {1} \\ \hline 0 & {0} \\ \hline {1} & {-1} \\ \hline {4} & {-2} \\ \hline {9} & {-3} \\ \hline\end{array}
$$

Linh V.
Numerade Educator

Problem 5

Nonlinear Functions
$$
y=x^{3}+2
$$

Linh V.
Numerade Educator

Problem 6

Nonlinear Functions
$$
y=\sqrt{x}
$$

Linh V.
Numerade Educator

Problem 7

Nonlinear Functions
$$
x=|y|
$$

Linh V.
Numerade Educator

Problem 8

Nonlinear Functions
$$
x=y^{2}+4
$$

Linh V.
Numerade Educator

Problem 9

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$y=2 x+3$$

Linh V.
Numerade Educator

Problem 10

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
y=-3 x+9
$$

Linh V.
Numerade Educator

Problem 11

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
2 y-x=5
$$

Linh V.
Numerade Educator

Problem 12

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
6 x-y=-1
$$

Linh V.
Numerade Educator

Problem 13

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
y=x(x+2)
$$

Linh V.
Numerade Educator

Problem 14

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
y=(x-2)(x+2)
$$

Linh V.
Numerade Educator

Problem 15

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
y=x^{2}
$$

Linh V.
Numerade Educator

Problem 16

List the ordered pairs obtained from each equation, given $1-2,$ $-1,0,1,2,3 \}$ as the domain. Graph each set of ordered pairs. Give the range.
$$
y=-4 x^{2}
$$

Linh V.
Numerade Educator

Problem 17

Give the domain of each function defined as follows.
$$
f(x)=2 x
$$

Linh V.
Numerade Educator

Problem 18

Give the domain of each function defined as follows.
$$
f(x)=2 x+3
$$

Linh V.
Numerade Educator

Problem 19

Give the domain of each function defined as follows.
$$
f(x)=x^{4}
$$

Linh V.
Numerade Educator

Problem 20

Give the domain of each function defined as follows.
$$
f(x)=(x+3)^{2}
$$

Linh V.
Numerade Educator

Problem 21

Give the domain of each function defined as follows.
$$
f(x)=\sqrt{4-x^{2}}
$$

Linh V.
Numerade Educator

Problem 22

Give the domain of each function defined as follows.
$$
f(x)=|3 x-6|
$$

Linh V.
Numerade Educator

Problem 23

Give the domain of each function defined as follows.
$$
f(x)=(x-3)^{1 / 2}
$$

Linh V.
Numerade Educator

Problem 24

Give the domain of each function defined as follows.
$$
f(x)=(3 x+5)^{1 / 2}
$$

Linh V.
Numerade Educator

Problem 25

Give the domain of each function defined as follows.
$$
f(x)=\frac{2}{1-x^{2}}
$$

Linh V.
Numerade Educator

Problem 26

Give the domain of each function defined as follows.
$$
f(x)=\frac{-8}{x^{2}-36}
$$

Linh V.
Numerade Educator

Problem 27

Give the domain of each function defined as follows.
$$
f(x)=-\sqrt{\frac{2}{x^{2}-16}}
$$

Linh V.
Numerade Educator

Problem 28

Give the domain of each function defined as follows.
$$
f(x)=-\sqrt{\frac{5}{x^{2}+36}}
$$

Linh V.
Numerade Educator

Problem 29

Give the domain of each function defined as follows.
$$
f(x)=\sqrt{x^{2}-4 x-5}
$$

Linh V.
Numerade Educator

Problem 30

Give the domain of each function defined as follows.
$$
f(x)=\sqrt{15 x^{2}+x-2}
$$

Linh V.
Numerade Educator

Problem 31

Give the domain of each function defined as follows.
$$
f(x)=\frac{1}{\sqrt{3 x^{2}+2 x-1}}
$$

Linh V.
Numerade Educator

Problem 32

Give the domain of each function defined as follows.
$$
f(x)=\sqrt{\frac{x^{2}}{3-x}}
$$

Linh V.
Numerade Educator

Problem 33

Give the domain and the range of each function. Where arrows are drawn, assume the function continues in the indicated direction.

Linh V.
Numerade Educator

Problem 34

Give the domain and the range of each function. Where arrows are drawn, assume the function continues in the indicated direction.

Linh V.
Numerade Educator

Problem 35

Give the domain and the range of each function. Where arrows are drawn, assume the function continues in the indicated direction.

Linh V.
Numerade Educator

Problem 36

Give the domain and the range of each function. Where arrows are drawn, assume the function continues in the indicated direction.

Linh V.
Numerade Educator

Problem 37

In Exercises $37-40,$ give the domain and range. Then, use each graph to find (a) $f(-2),(b) f(0),(c) f(1 / 2),$ and $(d)$ any values of $x$ such that $f(x)=1$

Linh V.
Numerade Educator

Problem 38

In Exercises $37-40,$ give the domain and range. Then, use each graph to find (a) $f(-2),(b) f(0),(c) f(1 / 2),$ and $(d)$ any values of $x$ such that $f(x)=1$

Linh V.
Numerade Educator

Problem 39

In Exercises $37-40,$ give the domain and range. Then, use each graph to find (a) $f(-2),(b) f(0),(c) f(1 / 2),$ and $(d)$ any values of $x$ such that $f(x)=1$

Linh V.
Numerade Educator

Problem 40

In Exercises $37-40,$ give the domain and range. Then, use each graph to find (a) $f(-2),(b) f(0),(c) f(1 / 2),$ and $(d)$ any values of $x$ such that $f(x)=1$

Linh V.
Numerade Educator

Problem 41

For each function, find (a) $f(4),(b) f(-1 / 2),(c) f(a),(d) f(2 / m),$ and $(e)$ any values of $x$ such that $f(x)=1$
$$
f(x)=3 x^{2}-4 x+1
$$

Linh V.
Numerade Educator

Problem 42

For each function, find (a) $f(4),(b) f(-1 / 2),(c) f(a),(d) f(2 / m),$ and $(e)$ any values of $x$ such that $f(x)=1$
$$
f(x)=(x+3)(x-4)
$$

Linh V.
Numerade Educator

Problem 43

For each function, find (a) $f(4),(b) f(-1 / 2),(c) f(a),(d) f(2 / m),$ and $(e)$ any values of $x$ such that $f(x)=1$
$$
f(x)=\left\{\begin{array}{ll}{\frac{2 x+1}{x-4}} & {\text { if } x \neq 4} \\ {7} & {\text { if } x=4}\end{array}\right.
$$

Linh V.
Numerade Educator

Problem 44

For each function, find (a) $f(4),(b) f(-1 / 2),(c) f(a),(d) f(2 / m),$ and $(e)$ any values of $x$ such that $f(x)=1$
$$
f(x)=\left\{\begin{array}{ll}{\frac{x-4}{2 x+1}} & {\text { if } x \neq-\frac{1}{2}} \\ {10} & {\text { if } x=-\frac{1}{2}}\end{array}\right.
$$

Linh V.
Numerade Educator

Problem 45

Let $f(x)=6 x^{2}-2$ and $g(x)=x^{2}-2 x+5$ to find the following values.
$$
f(t+1)
$$

Linh V.
Numerade Educator

Problem 46

Let $f(x)=6 x^{2}-2$ and $g(x)=x^{2}-2 x+5$ to find the following values.
$$
f(2-r)
$$

Linh V.
Numerade Educator

Problem 47

Let $f(x)=6 x^{2}-2$ and $g(x)=x^{2}-2 x+5$ to find the following values.
$$
g(r+h)
$$

Linh V.
Numerade Educator

Problem 48

Let $f(x)=6 x^{2}-2$ and $g(x)=x^{2}-2 x+5$ to find the following values.
$$
g(z-p)
$$

Linh V.
Numerade Educator

Problem 49

Let $f(x)=6 x^{2}-2$ and $g(x)=x^{2}-2 x+5$ to find the following values.
$$
g\left(\frac{3}{q}\right)
$$

Linh V.
Numerade Educator

Problem 50

Let $f(x)=6 x^{2}-2$ and $g(x)=x^{2}-2 x+5$ to find the following values.
$$
g\left(-\frac{5}{z}\right)
$$

Linh V.
Numerade Educator

Problem 51

For each function defined as follows, find $(\text { a }) f(x+h),$ (b) $f(x+h)-f(x),$ and (c) $[f(x+h)-f(x)]/h.$
$$
f(x)=2 x+1
$$

Linh V.
Numerade Educator

Problem 52

For each function defined as follows, find $(\text { a }) f(x+h),$ (b) $f(x+h)-f(x),$ and (c) $[f(x+h)-f(x)]/h.$
$$
f(x)=x^{2}-3
$$

Linh V.
Numerade Educator

Problem 53

For each function defined as follows, find $(\text { a }) f(x+h),$ (b) $f(x+h)-f(x),$ and (c) $[f(x+h)-f(x)]/h.$
$$
f(x)=2 x^{2}-4 x-5
$$

Linh V.
Numerade Educator

Problem 54

For each function defined as follows, find $(\text { a }) f(x+h),$ (b) $f(x+h)-f(x),$ and (c) $[f(x+h)-f(x)]/h.$
$$
f(x)=-4 x^{2}+3 x+2
$$

Linh V.
Numerade Educator

Problem 55

For each function defined as follows, find $(\text { a }) f(x+h),$ (b) $f(x+h)-f(x),$ and (c) $[f(x+h)-f(x)]/h.$
$$
f(x)=\frac{1}{x}
$$

Linh V.
Numerade Educator

Problem 56

For each function defined as follows, find $(\text { a }) f(x+h),$ (b) $f(x+h)-f(x),$ and (c) $[f(x+h)-f(x)]/h.$
$$
f(x)=-\frac{1}{x^{2}}
$$

Linh V.
Numerade Educator

Problem 57

Decide whether each graph represents a function.

Linh V.
Numerade Educator

Problem 58

Decide whether each graph represents a function.

Linh V.
Numerade Educator

Problem 59

Decide whether each graph represents a function.

Linh V.
Numerade Educator

Problem 60

Decide whether each graph represents a function.

Linh V.
Numerade Educator

Problem 61

Decide whether each graph represents a function.

Linh V.
Numerade Educator

Problem 62

Decide whether each graph represents a function.

Linh V.
Numerade Educator

Problem 63

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=3 x
$$

Linh V.
Numerade Educator

Problem 64

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=5 x
$$

Linh V.
Numerade Educator

Problem 65

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=2 x^{2}
$$

Linh V.
Numerade Educator

Problem 66

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=x^{2}-3
$$

Linh V.
Numerade Educator

Problem 67

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=\frac{1}{x^{2}+4}
$$

Linh V.
Numerade Educator

Problem 68

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=x^{3}+x
$$

Linh V.
Numerade Educator

Problem 69

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=\frac{x}{x^{2}-9}
$$

Linh V.
Numerade Educator

Problem 70

Classify each of the functions in Exercises 63–70 as even, odd, or neither.
$$
f(x)=|x-2|
$$

Linh V.
Numerade Educator

Problem 71

A chain-saw rental firm charges $\$ 28$ per day or fraction of a day to rent a saw, plus a fixed fee of $\$ 8$ for re sharpening the blade. Let $S(x)$ represent the cost of renting a saw for $x$ days. Find the following.
a. $S\left(\frac{1}{2}\right) \qquad$ b. $S(1) \qquad$ c. $S\left(1 \frac{1}{4}\right)$
d. $S\left(3 \frac{1}{2}\right) \quad$ e. $S(4) \qquad$ f. $S\left(4 \frac{1}{10}\right)$
g. What does it cost to rent a saw for 4$\frac{9}{10}$ days?
h. A portion of the graph of $y=S(x)$ is shown here. Explain how the graph could be continued.
i. What is the independent variable?
j. What is the dependent variable?
k. Is S a linear function? Explain.
l. Write a sentence or two explaining what part f and its answer represent.
m. We have left $x=0$ out of the graph. Discuss why it should or shouldn't be included. If it were included, how would you define $S(0) ?$

Check back soon!

Problem 71

A chain-saw rental firm charges $\$ 28$ per day or fraction of a day to rent a saw, plus a fixed fee of $\$ 8$ for re sharpening the blade. Let $S(x)$ represent the cost of renting a saw for $x$ days. Find the following.
a. $S\left(\frac{1}{2}\right) \qquad$ b. $S(1) \qquad$ c. $S\left(1 \frac{1}{4}\right)$
d. $S\left(3 \frac{1}{2}\right) \quad$ e. $S(4) \qquad$ f. $S\left(4 \frac{1}{10}\right)$

Linh V.
Numerade Educator

Problem 72

The cost to rent a mid-size car is $\$ 54$ per day or fraction of a day. If the car is picked up in Pittsburgh and dropped off in Cleveland, there is a fixed $\$ 44$ drop-off charge. Let $C(x)$ represent the cost of renting the car for $x$ days, taking it from Pittsburgh to Cleveland. Find the following.
$\begin{array}{llll}{\text { a.C(3/4) }} & {\text { b. } C(9 / 10)} & {\text { c. } C(1)} & {\text { d. } C\left(1 \frac{5}{8}\right)}\end{array}$
e. Find the cost of renting the car for 2.4 days.
f. Graph $y=C(x)$
g. Is $C$ a function? Explain.
h. Is $C$ a linear function? Explain.
i. What is the independent variable?
j. What is the dependent variable?

Check back soon!

Problem 73

According to Massachusetts state law, the maximum amount of a jury award that attorneys can receive is:
40$\%$ of the first $\$ 150,000$
33.3$\%$ of the next $\$ 150,000$
30$\%$ of the next $\$ 200,000,$ and
24$\%$ of anything over $\$ 500,000$ .
Let $f(x)$ represent the maximum amount of money that an attorney in Massachusetts can receive for a jury award of size $x$ . Find each of the following, and describe in a sentence what the answer tells you. Source: The New Yorker.
a. $f(250,000) \quad$ b. $f(350,000) \quad$ c. $f(550,000)$
d. Sketch a graph of $f(x)$

Check back soon!

Problem 74

In New York state in 2010, the income tax rates for a single person were as follows:
4$\%$ of the first $\$ 8000$ earned,
4.5$\%$ of the next $\$ 3000$ earned,
5.25$\%$ of the next $\$ 2000$ earned,
5.9$\%$ of the next $\$ 7000$ earned,
6.85$\%$ of the next $\$ 180,000$ earned,
7.85$\%$ of the next $\$ 300,000$ earned, and
8.97$\%$ of any amount earned over $\$ 500,000$ .
Let $f(x)$ represent the amount of tax owed on an income of $x$ dollars. Find each of the following, and explain in a sentence what the answer tells you. Source: New York State.
a. $f(10,000)$
b. $f(12,000)$
c. $f(18,000)$
d. Sketch a graph of $f(x)$

Check back soon!

Problem 75

The figure in the next column shows the depth of a diving sperm whale as a function of time, as recorded by researchers at the Woods Hole Oceanographic Institution in Massachusetts. Source: Peter Tyack, Woods Hole Oceano graphic Institution.
Find the depth of the whale at the following times.
a. 17 hours and 37 minutes
b. 17 hours and 39 minutes

Linh V.
Numerade Educator

Problem 76

Metabolic Rate The basal metabolic rate (in kcal/day) for large anteaters is given by
$$
y=f(x)=19.7 x^{0.753}
$$
where $x$ is the anteater's weight in kilograms. Source: Wildlife Feeding and Nutrition.
a. Find the basal metabolic rate for anteaters with the following weights.
i. 5 $\mathrm{kg} \quad$ ii. 25 $\mathrm{kg}$
b. Suppose the anteater's weight is given in pounds rather than kilograms. Given that $1 \mathrm{lb}=0.454 \mathrm{kg},$ find a function $x=g(z)$ giving the anteater's weight in kilograms if $z$ is the animal's weight in pounds.
c. Write the basal metabolic rate as a function of the weight in pounds in the form $y=a z^{b}$ by calculating $f(g(z)) .$

Check back soon!

Problem 77

The energy expenditure (in kcal/km) for animals swimming at the surface of the water is given by
$$y=f(x)=0.01 x^{0.88}$$
where $x$ is the animal's weight in grams. Source: Wildlife Feeding and Nutrition.
a. Find the energy for the following animals swimming at the surface of the water.
$$
\begin{array}{l}{\text { i. A muskrat weighing } 800 \mathrm{g}} \\ {\text { ii. A sea otter weighing } 20,000 \mathrm{g}}\end{array}
$$
b. Suppose the animal's weight is given in kilograms rather than grams. Given that $1 \mathrm{kg}=1000 \mathrm{g},$ find a function $x=g(z)$ giving the animal's weight in grams if $z$ is the animal's weight in kilograms.
c. Write the energy expenditure as a function of the weight in kilograms in the form $y=a z^{b}$ by calculating $f(g(z))$ .
*Technically, kilograms are a measure of mass, not weight. Weight is a measure of the force of gravity, which varies with the distance from the center of Earth. For objects on the surface of Earth, weight and mass are often used interchangeably, and we will do so in this text.

Check back soon!

Problem 78

Over the last century, the world has shifted from using high-carbon sources of energy such as wood to lower carbon fuels such as oil and natural gas, as shown in the figure. Source: The New York Times. The rise in carbon emissions during this time has caused concern because of its suspected contribution to global warming.
a. In what year were the percent of wood and coal use equal? What was the percent of each used in that year?
b. In what year were the percent of oil and coal use equal? What was the percent of each used in that year?

Check back soon!

Problem 79

A rectangular field is to have an area of 500 $\mathrm{m}^{2}$
a. Write the perimeter, $P,$ of the field as a function of the width, $w$
b. Find the domain of the function in part a.
c. Use a graphing calculator to sketch the graph of the function in part a.
d. Describe what the graph found in part c tells you about how the perimeter of the field varies with the width.

Check back soon!

Problem 80

A rectangular field is to have a perimeter of 6000 $\mathrm{ft}$ .
a. Write the area, $A,$ of the field as a function of the width, $w$
b. Find the domain of the function in part a.
c. Use a graphing calculator to sketch the graph of the function in part a.
d. Describe what the graph found in part c tells you about how the area of the field varies with the width.

Check back soon!