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Thomas Calculus

George B. Thomas Jr.

Chapter 1

Functions - all with Video Answers

Educators

+ 13 more educators

Section 1

Functions and Their Graphs

01:56

Problem 1

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

Sanchit Jain
Sanchit Jain
Numerade Educator
02:19

Problem 2

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

Mengchun Cai
Mengchun Cai
Numerade Educator
01:17

Problem 3

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

Aman Gupta
Aman Gupta
Numerade Educator
04:17

Problem 4

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:51

Problem 5

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

WZ
Wen Zheng
Numerade Educator
05:37

Problem 6

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:48

Problem 7

Which of the graphs are graphs of functions of $x,$ and which are not? Give reasons for your answers.

Chris Trentman
Chris Trentman
Numerade Educator
02:01

Problem 8

Which of the graphs are graphs of functions of $x,$ and which are not? Give reasons for your answers.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:47

Problem 9

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

Gregory Higby
Gregory Higby
Numerade Educator
03:13

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.

Karl Schaefer
Karl Schaefer
University of Chicago
06:03

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.

Linda Hand
Linda Hand
Numerade Educator
04:13

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.

Karl Schaefer
Karl Schaefer
University of Chicago
View

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 $x .$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:08

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 $y .$

Willis James
Willis James
Numerade Educator
00:58

Problem 15

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

WZ
Wen Zheng
Numerade Educator
02:43

Problem 16

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:37

Problem 17

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

WZ
Wen Zheng
Numerade Educator
02:54

Problem 18

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:33

Problem 19

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

WZ
Wen Zheng
Numerade Educator
03:30

Problem 20

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

Karl Schaefer
Karl Schaefer
University of Chicago
02:09

Problem 21

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

Sanchit Jain
Sanchit Jain
Numerade Educator
02:02

Problem 22

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:34

Problem 23

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

WZ
Wen Zheng
Numerade Educator
06:08

Problem 24

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:10

Problem 25

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

WZ
Wen Zheng
Numerade Educator
02:43

Problem 26

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:58

Problem 27

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

WZ
Wen Zheng
Numerade Educator
01:37

Problem 28

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

Karl Schaefer
Karl Schaefer
University of Chicago
03:11

Problem 29

Find a formula for each function graphed.

Chris Trentman
Chris Trentman
Numerade Educator
07:09

Problem 30

Find a formula for each function graphed.

Karl Schaefer
Karl Schaefer
University of Chicago
01:50

Problem 31

Find a formula for each function graphed.

WZ
Wen Zheng
Numerade Educator
07:06

Problem 32

Find a formula for each function graphed.

Karl Schaefer
Karl Schaefer
University of Chicago
01:20

Problem 33

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

WZ
Wen Zheng
Numerade Educator
03:31

Problem 34

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

Mengchun Cai
Mengchun Cai
Numerade Educator
01:32

Problem 35

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

WZ
Wen Zheng
Numerade Educator
02:07

Problem 36

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:34

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.

$y=-x^{3}$

WZ
Wen Zheng
Numerade Educator
02:21

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.

$y=-\frac{1}{x^{2}}$

Karl Schaefer
Karl Schaefer
University of Chicago
00:59

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.

$y=-\frac{1}{x}$

WZ
Wen Zheng
Numerade Educator
01:14

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.

$y=\frac{1}{|x|}$

Karl Schaefer
Karl Schaefer
University of Chicago
01:23

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.

$y=\sqrt{|x|}$

WZ
Wen Zheng
Numerade Educator
01:16

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.

$y=\sqrt{-x}$

Karl Schaefer
Karl Schaefer
University of Chicago
01:25

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.

$y=x^{3} / 8$

WZ
Wen Zheng
Numerade Educator
02:23

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.

$y=-4 \sqrt{x}$

Karl Schaefer
Karl Schaefer
University of Chicago
01:43

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.

$y=-x^{3 / 2}$

WZ
Wen Zheng
Numerade Educator
04:26

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.

$y=(-x)^{2 / 3}$

Karl Schaefer
Karl Schaefer
University of Chicago
00:36

Problem 47

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

WZ
Wen Zheng
Numerade Educator
01:31

Problem 48

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:38

Problem 49

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

WZ
Wen Zheng
Numerade Educator
01:57

Problem 50

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:48

Problem 51

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

WZ
Wen Zheng
Numerade Educator
01:16

Problem 52

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:38

Problem 53

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

WZ
Wen Zheng
Numerade Educator
01:43

Problem 54

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:57

Problem 55

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

WZ
Wen Zheng
Numerade Educator
01:31

Problem 56

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:48

Problem 57

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

WZ
Wen Zheng
Numerade Educator
01:19

Problem 58

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:52

Problem 59

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

WZ
Wen Zheng
Numerade Educator
01:35

Problem 60

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:43

Problem 61

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

WZ
Wen Zheng
Numerade Educator
01:38

Problem 62

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

Karl Schaefer
Karl Schaefer
University of Chicago
00:50

Problem 63

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

WZ
Wen Zheng
Numerade Educator
02:09

Problem 64

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 $v=10 \mathrm{m} / \mathrm{sec} ?$

Karl Schaefer
Karl Schaefer
University of Chicago
01:16

Problem 65

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
01:57

Problem 66

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:16

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 $x .$

Aman Gupta
Aman Gupta
Numerade Educator
06:42

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 $x .$

Karl Schaefer
Karl Schaefer
University of Chicago
00:48

Problem 69

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

WZ
Wen Zheng
Numerade Educator
02:14

Problem 70

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

Karl Schaefer
Karl Schaefer
University of Chicago
01:44

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
$$\frac{x}{2}>1+\frac{4}{x}$$
b. Confirm your findings in part (a) algebraically.

WZ
Wen Zheng
Numerade Educator
11:30

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.

Karl Schaefer
Karl Schaefer
University of Chicago
00:41

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 $y=0$ .

WZ
Wen Zheng
Numerade Educator
02:55

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.

Karl Schaefer
Karl Schaefer
University of Chicago
04:58

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

Leon Druch
Leon Druch
Numerade Educator
09:04

Problem 76

Industrial costs A power plant sits next to a river where the river is 800 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 if the least expensive location for point $Q$ is less than 2000 ft or greater than 2000 ft from point $P .$

Karl Schaefer
Karl Schaefer
University of Chicago