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

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

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

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

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

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

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

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

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

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

Give the domain of each function defined as follows.

$$

f(x)=2 x

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=2 x+3

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=x^{4}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=(x+3)^{2}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\sqrt{4-x^{2}}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=|3 x-6|

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=(x-3)^{1 / 2}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=(3 x+5)^{1 / 2}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\frac{2}{1-x^{2}}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\frac{-8}{x^{2}-36}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=-\sqrt{\frac{2}{x^{2}-16}}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=-\sqrt{\frac{5}{x^{2}+36}}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\sqrt{x^{2}-4 x-5}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\sqrt{15 x^{2}+x-2}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\frac{1}{\sqrt{3 x^{2}+2 x-1}}

$$

Linh V.

Numerade Educator

Give the domain of each function defined as follows.

$$

f(x)=\sqrt{\frac{x^{2}}{3-x}}

$$

Linh V.

Numerade Educator

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

Linh V.

Numerade Educator

Linh V.

Numerade Educator

Linh V.

Numerade Educator

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

Linh V.

Numerade Educator

Linh V.

Numerade Educator

Linh V.

Numerade Educator

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Classify each of the functions in Exercises 63–70 as even, odd, or neither.

$$

f(x)=3 x

$$

Linh V.

Numerade Educator

Classify each of the functions in Exercises 63–70 as even, odd, or neither.

$$

f(x)=5 x

$$

Linh V.

Numerade Educator

Classify each of the functions in Exercises 63–70 as even, odd, or neither.

$$

f(x)=2 x^{2}

$$

Linh V.

Numerade Educator

Classify each of the functions in Exercises 63–70 as even, odd, or neither.

$$

f(x)=x^{2}-3

$$

Linh V.

Numerade Educator

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

Classify each of the functions in Exercises 63–70 as even, odd, or neither.

$$

f(x)=x^{3}+x

$$

Linh V.

Numerade Educator

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

Classify each of the functions in Exercises 63–70 as even, odd, or neither.

$$

f(x)=|x-2|

$$

Linh V.

Numerade Educator

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 $S(0) ?$

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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

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?

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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 $f(x)$

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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 $f(x)$

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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

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 $f(g(z)) .$

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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.

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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?

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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.

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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.

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