Thomas Calculus

George B. Thomas, Jr.

Chapter 1

Functions - all with Video Answers

Educators

+ 20 more educators

Section 1

Functions and Their Graphs

Problem 1

In Exercises $1-6,$ find the domain and range of each function.
$$f(x)=1+x^{2}$$

Mengchun Cai

Mengchun Cai

Numerade Educator

Problem 2

In Exercises $1-6,$ find the domain and range of each function.
$$f(x)=1-\sqrt{x}$$

Willis James

Numerade Educator

Problem 3

In Exercises $1-6,$ find the domain and range of each function.
$$
F(x)=\sqrt{5 x+10}
$$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 4

In Exercises $1-6,$ find the domain and range of each function.
$$g(x)=\sqrt{x^{2}-3 x}$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 5

In Exercises $1-6,$ find the domain and range of each function.
$$f(t)=\frac{4}{3-t}$$

Carson Merrill

Numerade Educator

Problem 6

In Exercises $1-6,$ find the domain and range of each function.
$$G(t)=\frac{2}{t^{2}-16}$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 7

In Exercises 7 and $8,$ which of the graphs are graphs of functions of $x,$
and which are not? Give reasons for your answers.

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 8

In Exercises 7 and $8,$ which of the graphs are graphs of functions of $x,$
and which are not? Give reasons for your answers.

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 9

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

Stephen Hobbs

Numerade Educator

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.

Willis James

Numerade Educator

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.

Willis James

Numerade Educator

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.

Nathalie Luna

The University of Texas Rio Grande Valley

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 $x .$

Chandler Austin

Chandler Austin

Numerade Educator

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 $y .$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 15

Find the natural domain and graph the functions in Exercises $15-20 .$
$$f(x)=5-2 x$$

Abby Kennedy

Numerade Educator

Problem 16

Find the natural domain and graph the functions in Exercises $15-20 .$
$$f(x)=1-2 x-x^{2}$$

Donna Densmore

Numerade Educator

Problem 17

Find the natural domain and graph the functions in Exercises $15-20 .$
$$g(x)=\sqrt{|x|}$$

Carson Merrill

Numerade Educator

Problem 18

Find the natural domain and graph the functions in Exercises $15-20 .$
$$g(x)=\sqrt{-x}$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 19

Find the natural domain and graph the functions in Exercises $15-20 .$
$$F(t)=t /|t|$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 20

Find the natural domain and graph the functions in Exercises $15-20 .$
$$G(t)=1 /|t|$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 21

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

Amy Jiang

Numerade Educator

Problem 22

Find the range of $$y=2+\frac{x^{2}}{x^{2}+4}$$

Alessandro Canino

Alessandro Canino

Numerade Educator

Problem 23

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

Stephen Hobbs

Numerade Educator

Problem 24

Graph the following equations and explain why they are not
graphs of functions of $x .$
$$
|x|+|y|=1 \quad \text { b. }|x+y|=1
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 25

Graph the functions in Exercises $25-28$
$$
f(x)=\left\{\begin{array}{ll}{x,} & {0 \leq x \leq 1} \\ {2-x,} & {1 < x \leq 2}\end{array}\right.
$$

James Kiss

Numerade Educator

Problem 26

Graph the functions in Exercises $25-28$
$$
g(x)=\left\{\begin{array}{ll}{1-x,} & {0 \leq x \leq 1} \\ {2-x,} & {1 < x \leq 2}\end{array}\right.
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 27

Graph the functions in Exercises $25-28$
$$
F(x)=\left\{\begin{array}{ll}{4-x^{2},} & {x \leq 1} \\ {x^{2}+2 x,} & {x>1}\end{array}\right.
$$

James Kiss

Numerade Educator

Problem 28

Graph the functions in Exercises $25-28$
$$
G(x)=\left\{\begin{array}{ll}{1 / x,} & {x<0} \\ {x,} & {0 \leq x}\end{array}\right.
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 29

Find a formula for each function graphed in Exercises $29-32$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 30

Find a formula for each function graphed in Exercises $29-32$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 31

Find a formula for each function graphed in Exercises $29-32$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 32

Find a formula for each function graphed in Exercises $29-32$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 33

For what values of $x$ is
$$
\lfloor x\rfloor= 0 ? \quad \text { b. } | x \rceil=0 ?
$$

WZ

Wen Zheng

Numerade Educator

Problem 34

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

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 35

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

Dena Ballagh

Numerade Educator

Problem 36

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

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 37

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=-x^{3}
$$

Abdul Ibitoye

Numerade Educator

Problem 38

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=-\frac{1}{x^{2}}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 39

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=-\frac{1}{x}
$$

Gregory Higby

Numerade Educator

Problem 40

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=\frac{1}{|x|}
$$

Gregory Higby

Numerade Educator

Problem 41

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=\sqrt{|x|}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 42

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=\sqrt{-x}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 43

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=x^{3} / 8
$$

Abdul Ibitoye

Numerade Educator

Problem 44

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=-4 \sqrt{x}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 45

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=-x^{3 / 2}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 46

Graph the functions in Exercises $37-46 .$ What symmetries, if any, do
the graphs have? Specify the intervals over which the function is
increasing and the intervals where it is decreasing.
$$
y=(-x)^{2 / 3}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 47

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
f(x)=3
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 48

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
f(x)=x^{-5}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 49

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
f(x)=x^{2}+1
$$

Santana Scott

Numerade Educator

Problem 50

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
f(x)=x^{2}+x
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 51

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
g(x)=x^{3}+x
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 52

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
g(x)=x^{4}+3 x^{2}-1
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 53

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
g(x)=\frac{1}{x^{2}-1}
$$

John Irizar

Numerade Educator

Problem 54

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
g(x)=\frac{x}{x^{2}-1}
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 55

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
h(t)=\frac{1}{t-1}
$$

Chester Kyles Colita

Chester Kyles Colita

Numerade Educator

Problem 56

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
h(t)=\left|t^{3}\right|
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 57

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
h(t)=2 t+1
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 58

In Exercises $47-58,$ say whether the function is even, odd, or neither.
Give reasons for your answer.
$$
h(t)=2|t|+1
$$

Nathalie Luna

The University of Texas Rio Grande Valley

Problem 59

The variable $s$ is proportional to $t,$ and $s=25$ when $t=75 .$
Determine $t$ when $s=60$

Lisa Stryjewski

Lisa Stryjewski

Numerade Educator

Problem 60

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

Andrew Bassila

Numerade Educator

Problem 61

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

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 62

Boyle's Law 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 $P=14.7 \mathrm{b} / \mathrm{in}^{2}$
when $V=1000 \mathrm{in}^{3},$ then what is $V$ when $P=23.4 \mathrm{lb} / \mathrm{in}^{2} ?$

Eric Carlsen

Numerade Educator

Problem 63

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 $x .$

Aman Gupta

Numerade Educator

Problem 64

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

Eric Carlsen

Numerade Educator

Problem 65

In Exercises 65 and $66,$ match each equation with its graph. Do not
use a graphing device, and give reasons for your answer.
$$
\text { a. } y=x^{4} \quad \text { b. } y=x^{7} \quad \text { c. } y=x^{10}
$$

Carson Merrill

Numerade Educator

Problem 66

In Exercises 65 and $66,$ match each equation with its graph. Do not
use a graphing device, and give reasons for your answer.
$$
\text { a. }y=5 x \quad \text { b. } y=5^{x} \quad \text { c. } y=x^{5}
$$

Carson Merrill

Numerade Educator

Problem 67

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

AG

Ankit Gupta

Numerade Educator

Problem 68

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.

Karl Schaefer

University of Chicago

Problem 69

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 $y=0 .$

WZ

Wen Zheng

Numerade Educator

Problem 70

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.

Carson Merrill

Numerade Educator

Problem 71

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 $\$ 5 / f t$ for the legs and $\$ 10 / f t$ for the hypotenuse, write the
total cost $C$ of construction as a function of $h .$

Leon Druch

Numerade Educator

Problem 72

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 $\mathrm{mi}$ 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 if the least expensive
location for point $Q$ is less than 2000 ft or greater than 2000 $\mathrm{ft}$
from point $P .$

Stephen Hobbs

Numerade Educator