University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 1

Functions

Educators


Problem 1

Find the domain and range of each function.
$$f(x)=1+x^{2}$$

Evan L.
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Problem 2

Find the domain and range of each function.
$$f(x)=1-\sqrt{x}$$

Andrew B.
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Problem 3

Find the domain and range of each function.
$$F(x)=\sqrt{5 x+10}$$

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Problem 4

Find the domain and range of each function.
$$g(x)=\sqrt{x^{2}-3 x}$$

Andrew B.
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Problem 5

Find the domain and range of each function.
$$f(t)=\frac{4}{3-t}$$

Evan L.
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Problem 6

Find the domain and range of each function.
$$G(t)=\frac{2}{t^{2}-16}$$

Andrew B.
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Problem 7

Which of the graphs are graphs of functions of $x$ and which are not? Give reasons for your answers.
(GRAPH CAN'T COPY)

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Problem 8

Which of the graphs are graphs of functions of $x$ and which are not? Give reasons for your answers.
(GRAPH CAN'T COPY)

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Problem 9

Express the area and perimeter of an equilateral triangle as a function of the triangle's side length $x$.

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Problem 10

Express the side length of a square as a function of the length $d$ of the square's diagonal. Then express the area as a function of the diagonal length.

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Problem 11

Express the edge length of a cube as a function of the cube's diagonal length $d$. Then express the surface area and volume of the cube as a function of the diagonal length.

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Problem 12

A point $P$ in the first quadrant lies on the graph of the function $f(x)=\sqrt{x} .$ Express the coordinates of $P$ as functions of the slope of the line joining $P$ to the origin.

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Problem 13

Consider the point $(x, y)$ lying on the graph of the line $2 x+4 y=5$ Let $L$ be the distance from the point $(x, y)$ to the origin $(0,0) .$ Write $L$ as a function of $x$

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Problem 14

Consider the point $(x, y)$ lying on the graph of $y=\sqrt{x-3}$. Let L be the distance between the points $(x, y)$ and $(4,0) .$ Write $L$ as a function of $y$

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Problem 15

Find the natural domain and graph the functions.
$$f(x)=5-2 x$$

Evan L.
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Problem 16

Find the natural domain and graph the functions.
$$f(x)=1-2 x-x^{2}$$

Andrew B.
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Problem 17

Find the natural domain and graph the functions.
$$g(x)=\sqrt{|x|}$$

Evan L.
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Problem 18

Find the natural domain and graph the functions.
$$g(x)=\sqrt{-x}$$

Andrew B.
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Problem 19

Find the natural domain and graph the functions.
$$F(t)=t /|t|$$

Evan L.
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Problem 20

Find the natural domain and graph the functions.
$$G(t)=1 /|t|$$

Andrew B.
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Problem 21

Find the domain of $y=\frac{x+3}{4-\sqrt{x^{2}-9}}$

Evan L.
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Problem 22

Find the range of $y=2+\sqrt{9+x^{2}}$

Andrew B.
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Problem 23

Graph the following equations and explain why they are not graphs of functions of $x$
a. $|y|=x$
b. $y^{2}=x^{2}$

Evan L.
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Problem 24

Graph the following equations and explain why they are not graphs of functions of $x$
a. $|x|+|y|=1$
b. $|x+y|=1$

Andrew B.
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Problem 25

Graph the functions.
$$f(x)=\left\{\begin{array}{ll}
x, & 0 \leq x \leq 1 \\
2-x, & 1<x \leq 2
\end{array}\right.$$

Evan L.
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Problem 26

Graph the functions.
$$g(x)=\left\{\begin{array}{ll}
1-x, & 0 \leq x \leq 1 \\
2-x, & 1<x \leq 2
\end{array}\right.$$

Andrew B.
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Problem 27

Graph the functions.
$$F(x)=\left\{\begin{array}{ll}
4-x^{2}, & x \leq 1 \\
x^{2}+2 x, & x>1
\end{array}\right.$$

Evan L.
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Problem 28

Graph the functions.
$$G(x)=\left\{\begin{array}{ll}
1 / x, & x<0 \\
x, & 0 \leq x
\end{array}\right.$$

Andrew B.
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Problem 29

Find a formula for each function graphed.
(GRAPH CAN'T COPY)

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Problem 30

Find a formula for each function graphed.
(GRAPH CAN'T COPY)

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Problem 31

Find a formula for each function graphed.
(GRAPH CAN'T COPY)

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Problem 32

Find a formula for each function graphed.
(GRAPH CAN'T COPY)

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Problem 33

For what values of $x$ is
a. $|x|=0 ?$
b. $\lceil x\rceil=0 ?$

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Problem 34

What real numbers $x$ satisfy the equation $\lfloor x\rfloor=\lceil x | ?$

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Problem 35

Does $\lceil-x\rceil=-\lfloor x\rfloor$ for all real $x ?$ Give reasons for your answer.

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Problem 36

Graph the function
$$
f(x)=\left\{\begin{array}{ll}
\lfloor x\rfloor, & x \geq 0 \\
| x\rceil, & x<0
\end{array}\right.
$$
Why is $f(x)$ called the integer part of $x ?$

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Problem 37

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=-x^{3}$$

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Problem 38

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=-\frac{1}{x^{2}}$$

Andrew B.
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Problem 39

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=-\frac{1}{x}$$

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Problem 40

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=\frac{1}{|x|}$$

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Problem 41

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=\sqrt{|x|}$$

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Problem 42

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=\sqrt{-x}$$

Andrew B.
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Problem 43

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=x^{3} / 8$$

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Problem 44

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=-4 \sqrt{x}$$

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Problem 45

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=-x^{3 / 2}$$

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Problem 46

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
$$y=(-x)^{2 / 3}$$

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Problem 47

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$f(x)=3$$

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Problem 48

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$f(x)=x^{-5}$$

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Problem 49

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$f(x)=x^{2}+1$$

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Problem 50

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$f(x)=x^{2}+x$$

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Problem 51

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$g(x)=x^{3}+x$$

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Problem 52

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$g(x)=x^{4}+3 x^{2}-1$$

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Problem 53

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$g(x)=\frac{1}{x^{2}-1}$$

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Problem 54

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$g(x)=\frac{x}{x^{2}-1}$$

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Problem 55

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$h(t)=\frac{1}{t-1}$$

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Problem 56

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$h(t)=\left|t^{3}\right|$$

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Problem 57

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$h(t)=2 t+1$$

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Problem 58

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$h(t)=2|t|+1$$

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Problem 59

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$\sin 2 x$$

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Problem 60

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$\sin x^{2}$$

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Problem 61

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$\cos 3 x$$

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Problem 62

Say whether the function is even, odd, or neither. Give reasons for your answer.
$$1+\cos x$$

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Problem 63

The variable $s$ is proportional to $t$, and $s=25$ when $t=75 .$ Determine $t$ when $s=69$

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Problem 64

The kinetic energy $K$ of a mass is proportional to the square of its velocity $v .$ If $K=12,960$ joules when $v=18 \mathrm{m} / \mathrm{scc},$ what is $K$ when $v=10 \mathrm{m} / \mathrm{sec} ?$

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Problem 65

The variables $r$ and $s$ are inversely proportional, and $r=6$ when $s=4 .$ Determine $s$ when $r=10$

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Problem 66

Boyle's Law says that the volume $V$ of a gas at constant temperature increases whenever the pressure $P$ decreases, so that $V$ and $P$ are inversely proportional. If $P=14.7 \mathrm{lb} / \mathrm{in}^{2}$ when $V=1000 \mathrm{in}^{3},$ then what is $V$ when $P=23.4 \mathrm{lb} / \mathrm{in}^{2} ?$

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Problem 67

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side $x$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $x$.
(FIGURE CAN'T COPY)

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Problem 68

The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.
a. Express the $y$ -coordinate of $P$ in terms of $x$. (You might start by writing an equation for the line $A B .$ )
b. Express the area of the rectangle in terms of $x$.

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Problem 69

Match each equation with its graph. Do not use a graphing device, and give reasons for your answer.
(GRAPH CAN'T COPY)

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Problem 70

Match each equation with its graph. Do not use a graphing device, and give reasons for your answer.
(GRAPH CAN'T COPY)

Andrew B.
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Problem 71

a. Graph the functions $f(x)=x / 2$ and $g(x)=1+(4 / x)$ together to identify the values of $x$ for which
$$
\frac{\sqrt{3}}{2}+1 \frac{4}{4}
$$
b. Confirm your findings in part (a) algebraically.

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Problem 72

a. Graph the functions $f(x)=3 /(x-1)$ and $g(x)=2 /(x+1)$ together to identify the values of $x$ for which
$$
\frac{3}{x-1}<\frac{2}{x+1}
$$
b. Confirm your findings in part (a) algebraically.

Andrew B.
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Problem 73

For a curve to be symmetric about the $x$ -axis, the point $(x, y)$ must lie on the curve if and only if the point $(x,-y)$ lies on the curve. Explain why a curve that is symmetric about the $x$ -axis is not the graph of a function, unless the function is $y=0$

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Problem 74

Three hundred books sell for $\$ 40$ each, resulting in a revenue of $(300)(\$ 40)=\$ 12,000 .$ For each $\$ 5$ increase in the price, 25 fewer books are sold. Write the revenue $R$ as a function of the number $x$ of $\$ 5$ increases.

Andrew B.
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Problem 75

A pen in the shape of an isosceles right triangle with legs of length $x$ ft and hypotenuse of length $h$ ft is to be built. If fencing costs S5/ft for the legs and $\$ 10 /$ ft for the hypotenuse, write the total cost $C$ of construction as a function of $h$

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Problem 76

Industrial costs A power plant sits next to a river where the river is $800 \mathrm{ft}$ wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $\$ 180$ per foot across the river and $\$ 100$ per foot along the land.a. Suppose that the cable goes from the plant to a point $Q$ on the opposite side that is $x$ ft from the point $P$ directly opposite the plant. Write a function $C(x)$ that gives the cost of laying the cable in terms of the distance $x$.
b. Generate a table of values to determine whether the least expensive location for point $Q$ is less than 2000 ft or greater than $2000 \mathrm{ft}$ from point $P$

Andrew B.
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