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Calculus

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 1

Functions - all with Video Answers

Educators

+ 45 more educators

Section 1

Functions and Their Graphs

01:29

Problem 1

In Exercises $1-6,$ find the domain and range of each function.

$$f(x)=1+x^{2}$$

Christy Galilei
Christy Galilei
Numerade Educator
01:43

Problem 2

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

Madi Sousa
Madi Sousa
Numerade Educator
04:39

Problem 3

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

DM
Darby Martin
Numerade Educator
04:55

Problem 4

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

Peter Moon
Peter Moon
Numerade Educator
05:05

Problem 5

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

Justin Sekerak
Justin Sekerak
Numerade Educator
07:56

Problem 6

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

Noah Collingwood
Noah Collingwood
Numerade Educator
06:41

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.

Yaw Asomani
Yaw Asomani
Numerade Educator
06:41

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.

Yaw Asomani
Yaw Asomani
Numerade Educator
01:49

Problem 9

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

Stephen Hobbs
Stephen Hobbs
Numerade Educator
05:36

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.

Matthew Winsor
Matthew Winsor
Numerade Educator
03:05

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.

Gregory Higby
Gregory Higby
Numerade Educator
02:15

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.

Suzanne W.
Suzanne W.
Numerade Educator
02:25

Problem 13

Consider the point $(x, y)$ lying on the graph of the line 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 $x$

Pawan Yadav
Pawan Yadav
Numerade Educator
06:05

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
03:11

Problem 15

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

Kyle Fang
Kyle Fang
Numerade Educator
00:34

Problem 16

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

Linh Vu
Linh Vu
Numerade Educator
04:14

Problem 17

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

Sara Sasani
Sara Sasani
Numerade Educator
00:31

Problem 18

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

Linh Vu
Linh Vu
Numerade Educator
04:40

Problem 19

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

Sarah Davis
Sarah Davis
Numerade Educator
00:37

Problem 20

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

Linh Vu
Linh Vu
Numerade Educator
05:21

Problem 21

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

Michael Fralick
Michael Fralick
Numerade Educator
03:22

Problem 22

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

SP
Sudheer Pimputkar
Numerade Educator
07:34

Problem 23

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

Ines Muro Contreras
Ines Muro Contreras
Numerade Educator
01:53

Problem 24

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

James Kiss
James Kiss
Numerade Educator
02:04

Problem 25

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

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
03:29

Problem 26

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

Chinmai Managoli
Chinmai Managoli
Numerade Educator
01:05

Problem 27

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

Carson Merrill
Carson Merrill
Numerade Educator
01:40

Problem 28

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

Andrew Bassila
Andrew Bassila
Numerade Educator
01:53

Problem 29

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

Stephen Hobbs
Stephen Hobbs
Numerade Educator
01:53

Problem 30

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

Stephen Hobbs
Stephen Hobbs
Numerade Educator
24:55

Problem 31

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

OT
Omar Twahir
Numerade Educator
01:53

Problem 32

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

Stephen Hobbs
Stephen Hobbs
Numerade Educator
View

Problem 33

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

James Kiss
James Kiss
Numerade Educator
02:16

Problem 34

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

Alexandra Embry
Alexandra Embry
Numerade Educator
04:31

Problem 35

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

Mengchun Cai
Mengchun Cai
Numerade Educator
02:08

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 $x ?$

Andrew Bassila
Andrew Bassila
Numerade Educator
04:01

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 in- creasing and the intervals where it is decreasing.
$$y=-x^{3}$$

FK
Fatih Kilinc
Numerade Educator
00:38

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 in- creasing and the intervals where it is decreasing.
$$y=-\frac{1}{x^{2}}$$

Linh Vu
Linh Vu
Numerade Educator
00:36

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 in- creasing and the intervals where it is decreasing.
$$y=-\frac{1}{x}$$

Linh Vu
Linh Vu
Numerade Educator
02:42

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

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
15:25

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.
$$y=\sqrt{|x|}$$

bb
Bcrypt_Sha256$$2B$12$3Uvb/3Qomeocg1Cb./Ie4.Ovr/Vz610Anhtv0B/9Zz4Ciy8Gpf2E2 Bcrypt_Sha256$$2B$12$3Uvb/3Qomeocg1Cb./Ie4.Vxv2Zhbpuklg39Jgk1Gvqlmx2F/A3Qy
Numerade Educator
07:19

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.
$$y=\sqrt{-x}$$

AP
Angela Poling
Numerade Educator
00:32

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.
$$y=x^{3} / 8$$

Linh Vu
Linh Vu
Numerade Educator
04:40

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.
$$y=-4 \sqrt{x}$$

LS
Lukas Sotil
Numerade Educator
00:33

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

Linh Vu
Linh Vu
Numerade Educator
08:10

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

Howard Francis
Howard Francis
Numerade Educator
00:34

Problem 47

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

Linh Vu
Linh Vu
Numerade Educator
03:57

Problem 48

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

Dominik Walle
Dominik Walle
Numerade Educator
06:50

Problem 49

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

Daniel Throop
Daniel Throop
Numerade Educator
03:57

Problem 50

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

Dominik Walle
Dominik Walle
Numerade Educator
00:43

Problem 51

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

Linh Vu
Linh Vu
Numerade Educator
05:15

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
Nathalie Luna
The University of Texas Rio Grande Valley
05:15

Problem 53

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

Nathalie Luna
Nathalie Luna
The University of Texas Rio Grande Valley
01:48

Problem 54

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

John Irizar
John Irizar
Numerade Educator
01:01

Problem 55

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

Linh Vu
Linh Vu
Numerade Educator
04:07

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
Nathalie Luna
The University of Texas Rio Grande Valley
03:45

Problem 57

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

Chester Kyles Colita
Chester Kyles Colita
Numerade Educator
03:45

Problem 58

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

Chester Kyles Colita
Chester Kyles Colita
Numerade Educator
05:37

Problem 59

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

Manan Sheel
Manan Sheel
Numerade Educator
03:25

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

Siena Cizdziel
Siena Cizdziel
Numerade Educator
01:16

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
01:38

Problem 62

Boyle's Law Boyle's Law says that the volume $V$ of a gas at con- stant temperature increases whenever the pressure $P$ decreases, so that $V$ and $P$ are inversely proportional. If $P=14.7 \mathrm{lbs} / \mathrm{in}^{2}$ when $V=1000 \mathrm{in}^{3},$ then what is $V$ when $P=23.4 \mathrm{lbs} / \mathrm{in}^{22}$ ?

Carson Merrill
Carson Merrill
Numerade Educator
05:33

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 sides as in the figure. Express the volume $V$ of the box as a function of $r$

RE
Robert Elzinga
Numerade Educator
04:28

Problem 64

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 $x .$

Andrew Bassila
Andrew Bassila
Numerade Educator
01:23

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.
a. $y=x^{4}$
b. $y=x^{7}$
c. $y=x^{10}$

Carson Merrill
Carson Merrill
Numerade Educator
01:25

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.
a. $y=5 x$
b. $y=5^{x}$
c. $y=x^{5}$

Carson Merrill
Carson Merrill
Numerade Educator
01:14

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
07:27

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.

Andrew Bassila
Andrew Bassila
Numerade Educator
00:41

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
02:05

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.

Sia Mittal
Sia Mittal
Numerade Educator
02:17

Problem 71

A pen in the shape of an isosceles right triangle with length $x \mathrm{ft}$ and hypotenuse of length $h \mathrm{ft}$ is to be built. If fencing costs $\$ 5 / \mathrm{ft}$ for the legs and $\$ 10 / \mathrm{ft}$ for the hypotenuse, write the total cost Cof construction as a function of $h$ .

Carson Merrill
Carson Merrill
Numerade Educator
10:06

Problem 72

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 $x$ 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 point $P.$

Andrew Bassila
Andrew Bassila
Numerade Educator