Preview Extract

INSTRUCTORโS SOLUTIONS
MANUAL
CALCULUS
AND ITS APPLICATIONS
BRIEF VERSION
TWELFTH EDITION
AND
CALCULUS
AND ITS APPLICATIONS
SECOND EDITION
Marvin L. Bittinger
Indiana University Purdue University Indianapolis
David J. Ellenbogen
Community College of Vermont
Scott A. Surgent
Arizona State University
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright ยฉ 2020, 2015 by Pearson Education, Inc.
Publishing as Pearson, 501 Boylston Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in
any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-13-518244-4
ISBN-10: 0-13-518244-1
Contents
Chapter R: Functions, Graphs, and Models …………………………………………………… 1
Chapter R Test …………………………………………………………………………………… 86
Chapter 1: Differentiation …………………………………………………………………………. 90
Chapter 1 Test…………………………………………………………………………………… 235
Chapter 2: Exponential and Logarithmic Functions ……………………………………. 239
Chapter 2 Test ………………………………………………………………………………….. 314
Chapter 3: Applications of Differentiation ………………………………………………… 318
Chapter 3 Test ………………………………………………………………………………….. 510
Chapter 4: Integration …………………………………………………………………………….. 527
Chapter 4 Test ………………………………………………………………………………….. 658
Chapter 5: Applications of Integration ……………………………………………………… 664
Chapter 5 Test ………………………………………………………………………………….. 763
Chapter 6: Functions of Several Variables ………………………………………………… 772
Chapter 6 Test ………………………………………………………………………………….. 895
Chapter 7: Trigonometric Functions ………………………………………………………… 901
Chapter 7 Test ………………………………………………………………………………….. 944
Chapter 8: Differential Equations …………………………………………………………….. 949
Chapter 8 Test ………………………………………………………………………………… 1015
Chapter 9: Sequences and Series ……………………………………………………………. 1021
Chapter 9 Test ………………………………………………………………………………… 1104
Chapter 10: Probability Distributions ……………………………………………………… 1109
Chapter 10 Test ………………………………………………………………………………. 1160
Chapter 11: Systems and Matrices (online at bit.ly/2ScEtHt) ……………………… 1165
Chapter 11 Test……………………………………………………………………………….. 1263
Chapter 12: Combinatorics and Probability (online at bit.ly/2HyJiH0) ………… 1267
Chapter 12 Test……………………………………………………………………………….. 1349
iii
Copyright ยฉ 2020 Pearson Education, Inc.
iv
Copyright ยฉ 2016 Pearson Education, Inc.
Chapter R
Functions, Graphs, and Models
Exercise Set R.1
1.
2.
3.
4.
x
y
๏ญ1
3
0
0
2
๏ญ6
Graph y ๏ฝ x ๏ญ 1 .
Graph y ๏ฝ x ๏ซ 4 .
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
x
y
๏ญ2
2
0
4
3
7
๏จ x, y ๏ฉ
๏จ๏ญ2, 2๏ฉ
๏จ0, 4๏ฉ
๏จ3, 7๏ฉ
1
Graph y ๏ฝ ๏ญ x .
4
๏จ x, y ๏ฉ
๏จ๏ญ1,3๏ฉ
๏จ0, 0๏ฉ
๏จ2, ๏ญ6๏ฉ
5.
5
Graph y ๏ฝ ๏ญ x ๏ซ 3 .
3
6.
Graph y ๏ฝ
2
x๏ญ4.
3
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
y
x
๏จ x, y ๏ฉ
๏ญ3
๏ญ6
0
๏ญ4
3
๏ญ2
Graph y ๏ฝ ๏ญ3x .
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
Copyright ยฉ 2020 Pearson Education, Inc.
๏จ๏ญ3, ๏ญ6๏ฉ
๏จ0, ๏ญ4๏ฉ
๏จ3, ๏ญ2๏ฉ
2
7.
Chapter R Functions, Graphs, and Models
Graph x ๏ซ y ๏ฝ 5 .
We solve for y first.
x๏ซ y ๏ฝ5
y ๏ฝ 5๏ญ x
subtract x from both sides
y ๏ฝ ๏ญx ๏ซ 5
commutative property
Next, we choose some x-values and calculate
the corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
x
y
๏ญ1
6
0
5
2
3
Next, we choose some x-values and calculate
the corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
x
y
๏ญ2
0
2
1
6
2
๏จ x, y ๏ฉ
๏จ๏ญ2, 0๏ฉ
๏จ2,1๏ฉ
๏จ6, 2๏ฉ
๏จ x, y ๏ฉ
๏จ๏ญ1, 6๏ฉ
๏จ0,5๏ฉ
๏จ2,3๏ฉ
11. Graph 2 x ๏ซ 5 y ๏ฝ 10 .
8.
Graph x ๏ญ y ๏ฝ 4 .
12. Graph 5 x ๏ญ 6 y ๏ฝ 12 .
We solve for y first.
5 x ๏ญ 6 y ๏ฝ 12
9.
subtract 5 x from both sides
๏ญ6 y ๏ฝ 12 ๏ญ 5 x
1
y๏ฝ
divide both sides by ๏ญ 6
๏จ12 ๏ญ 5 x ๏ฉ
๏ญ6
5
y ๏ฝ ๏ญ2 ๏ซ x
6
5
y ๏ฝ x๏ญ2
6
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
Graph 6 x ๏ซ 3 y ๏ฝ ๏ญ9 .
10. Graph 8 y ๏ญ 2 x ๏ฝ 4 .
We solve for y first.
8 y ๏ญ 2x ๏ฝ 4
8 y ๏ฝ 2x ๏ซ 4
2
4
x๏ซ
8
8
1
1
y ๏ฝ x๏ซ
4
2
y๏ฝ
add 2x to both sides
divide both sides by 8
x
y
๏ญ6
๏ญ7
0
๏ญ2
6
3
Copyright ยฉ 2020 Pearson Education, Inc.
๏จ x, y ๏ฉ
๏จ๏ญ6, 7๏ฉ
๏จ0, ๏ญ2๏ฉ
๏จ6,3๏ฉ
Exercise Set R.1
3
16. Graph x ๏ฝ 2 ๏ญ y 2 .
Since x is expressed in terms of y we first
choose values for y and then compute x. Then
we plot the points that are found and connect
them with a smooth curve.
13. Graph y ๏ฝ x 2 ๏ญ 5 .
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
x
y
๏จ x, y ๏ฉ
๏ญ2
๏ญ1
๏ญ1
๏ญ4
0
๏ญ5
1
๏ญ4
2
๏ญ1
๏จ๏ญ2, ๏ญ1๏ฉ
๏จ๏ญ1, ๏ญ4๏ฉ
๏จ0, ๏ญ5๏ฉ
๏จ1, ๏ญ4๏ฉ
๏จ2, ๏ญ1๏ฉ
x
y
๏จ x, y ๏ฉ
๏ญ2
๏ญ2
1
๏ญ1
2
0
๏ญ1
1
๏ญ2
2
๏จ๏ญ2, ๏ญ2๏ฉ
๏จ1, ๏ญ1๏ฉ
๏จ2, 0๏ฉ
๏จ๏ญ1,1๏ฉ
๏จ๏ญ2, 2๏ฉ
17. Graph y ๏ฝ 5 .
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
14. Graph y ๏ฝ x 2 ๏ญ 3 .
x
y
๏ญ2
5
๏ญ1
5
0
5
1
5
2
5
15. Graph x ๏ฝ y 2 ๏ซ 2 .
Copyright ยฉ 2020 Pearson Education, Inc.
๏จ x, y ๏ฉ
๏จ๏ญ2,5๏ฉ
๏จ๏ญ1,5๏ฉ
๏จ0,5๏ฉ
๏จ1,5๏ฉ
๏จ2,5๏ฉ
4
Chapter R Functions, Graphs, and Models
18. Graph y ๏ฝ ๏ญ2 .
22. Graph y ๏ซ 1 ๏ฝ x3 .
First we solve for y.
y ๏ซ 1 ๏ฝ x3
y ๏ฝ x3 ๏ญ 1
Next, we choose some x-values and calculate
the corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
19. Graph y ๏ฝ 7 ๏ญ x 2 .
We choose some x-values and calculate the
corresponding y-values to find some ordered
pairs that are solutions of the equation. Then
we plot the points and connect them with a
smooth curve.
x
y
๏ญ2
3
๏ญ1
6
0
7
1
6
2
3
๏จ x, y ๏ฉ
๏จ๏ญ2,3๏ฉ
๏จ๏ญ1, 6๏ฉ
๏จ0, 7๏ฉ
๏จ1,6๏ฉ
๏จ2,3๏ฉ
23.
y
๏จ x, y ๏ฉ
๏ญ2
๏ญ9
๏ญ1
๏ญ2
0
๏ญ1
1
0
2
7
๏จ๏ญ2, ๏ญ9๏ฉ
๏จ๏ญ1, ๏ญ2๏ฉ
๏จ0, ๏ญ1๏ฉ
๏จ1,0๏ฉ
๏จ2, 7๏ฉ
A ๏ฝ 0.5t 4 ๏ซ 3.45t 3 ๏ญ 96.65t 2 ๏ซ 347.7t ,
0๏ฃt๏ฃ6
We substitute t ๏ฝ 2
A ๏ฝ 0.5 ๏จ 2๏ฉ ๏ซ 3.45 ๏จ 2๏ฉ ๏ญ 96.65 ๏จ 2๏ฉ ๏ซ 347.7 ๏จ 2๏ฉ
4
20. Graph y ๏ฝ 5 ๏ญ x 2 .
3
2
๏ฝ 344.4
Approximately 344.4 milligrams of ibuprofen
will remain in the blood stream 2 hours after
400 mg have been swallowed.
24.
21. Graph y ๏ญ 7 ๏ฝ x3 .
x
R ๏ฝ ๏ญ0.006 x ๏ซ 15.714
We substitute 1954 in for x to get
R ๏ฝ ๏ญ0.006 ๏จ1954๏ฉ ๏ซ 15.714
๏ฝ 3.99
According to this model, the world record for
the mile in 1954 is approximately 3.99 minutes.
Likewise, we substitute 2000 in for x to get
R ๏ฝ ๏ญ0.006 ๏จ 2000๏ฉ ๏ซ 15.714
๏ฝ 3.71
According to the model, the world record for
the mile in 2000 will be approximately 3.71
minutes.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.1
5
Finally, we substitute 2025 in for x to get
R ๏ฝ ๏ญ0.006 ๏จ 2025๏ฉ ๏ซ 15.714
๏ฝ 3.56
According to the model, the world record for
the mile in 2025 will be approximately 3.56
minutes.
27. a. N ๏ฝ 0.0319t ๏ซ 3.081
We substitute t ๏ฝ 13
N ๏ฝ 0.0319(13) ๏ซ 3.081
๏ฝ 3.46
In 2020, the number of female athletes will be
approximately 3.46 million.
b. N ๏ฝ 0.0319t ๏ซ 3.081
We substitute N ๏ฝ 3.6
3.6 ๏ฝ 0.0319t ๏ซ 3.081
t ๏ป 17.7
In the year 2007 + 17 = 2024, the number of
female athletes will be approximately 3.6
million.
25. a. A ๏ฝ ๏ญ0.002 x 2 ๏ซ 0.924 x ๏ญ 0.152
We substitute x ๏ฝ 21.3
A ๏ฝ ๏ญ0.002(21.3) 2 ๏ซ 0.924(21.3) ๏ญ 0.152
๏ฝ 18.62
The optimum angle, in degrees, to tilt the solar
panel in Honolulu is 18.62.
b. A ๏ฝ ๏ญ0.002 x 2 ๏ซ 0.924 x ๏ญ 0.152
We substitute x ๏ฝ 39.1
A ๏ฝ ๏ญ0.002(39.1) 2 ๏ซ 0.924(39.1) ๏ญ 0.152
๏ฝ 32.919
The optimum angle, in degrees, to tilt the solar
panel in Kansas City is 32.919.
28. a. N ๏ฝ ๏ญ0.0011t 2 ๏ซ 0.0412t ๏ซ 3.032
We substitute t ๏ฝ 13
N ๏ฝ ๏ญ0.0011(13) 2 ๏ซ 0.0412(13) ๏ซ 3.032
๏ป 3.38
According to this model, in 2020 there will be
approximately 3.38 million female athletes.
This model predicts a slightly lower level than
the model in exercise 25.
c. A ๏ฝ ๏ญ0.002 x 2 ๏ซ 0.924 x ๏ญ 0.152
We substitute x ๏ฝ 53.5
A ๏ฝ ๏ญ0.002(53.5)2 ๏ซ 0.924(53.5) ๏ญ 0.152
๏ฝ 43.558
The optimum angle, in degrees, to tilt the solar
panel in Edmonton is 43.558.
b. N ๏ฝ ๏ญ0.0011t 2 ๏ซ 0.0412t ๏ซ 3.032
We substitute t ๏ฝ 30
N ๏ฝ ๏ญ0.0011(30) 2 ๏ซ 0.0412(30) ๏ซ 3.032
๏ป 3.28
According to this model, in 2037 there will be
approximately 3.28 million female athletes.
26. a. S ๏ฝ ๏ญ0.00173t 2 ๏ซ 3.477t ๏ซ 0.924
We substitute t ๏ฝ 10
S ๏ฝ ๏ญ0.00173(10) 2 ๏ซ 3.477(10) ๏ซ 0.924
๏ฝ 35.5
In 2003, the sea had risen 35.5 mm over the
1993 level.
b. S ๏ฝ ๏ญ0.00173t 2 ๏ซ 3.477t ๏ซ 0.924
We substitute t ๏ฝ 27
S ๏ฝ ๏ญ0.00173(27) 2 ๏ซ 3.477(27) ๏ซ 0.924
๏ป 93.5
In 2020, the sea will rise approximately 93.5
mm over the 1993 level.
c. S ๏ฝ ๏ญ0.00173t 2 ๏ซ 3.477t ๏ซ 0.924
We substitute S ๏ฝ 1000
2
1000 ๏ฝ ๏ญ0.00173t ๏ซ 3.477t ๏ซ 0.924
t ๏ป 93.5
In 2024, the sea level will have risen 1 meter
above the 1993 level.
c. Answer will vary.
29.
v ๏จt ๏ฉ ๏ฝ 10.9t
We substitute 2.5 in for t to get
v ๏จ2.5๏ฉ ๏ฝ 10.9 ๏จ 2.5๏ฉ
๏ฝ 27.25
White was traveling at 27.25 miles per hour
when he reentered the half pipe.
30.
s ๏จt ๏ฉ ๏ฝ 16t 2
s ๏จt ๏ฉ ๏ฝ 28
28 ๏ฝ 16t 2
28 2
๏ฝt
16
28
๏ฝ t2
16
1.3228 ๏ป t
Copyright ยฉ 2020 Pearson Education, Inc.
6
Chapter R Functions, Graphs, and Models
Danny took approximately 1.32 seconds to hit
the ramp.
31. a) A ๏ฝ P ๏จ1 ๏ซ i ๏ฉ
t
1
๏ฝ 100, 000 ๏จ1.028๏ฉ
nt
๏ฝ 100, 000 ๏จ1 ๏ซ 0.00000196347 ๏ฉ
๏ฝ 100, 000 ๏จ1.00000196347 ๏ฉ
8760
๏ฝ 100, 000 ๏จ1.02839563811๏ฉ
๏ป 102,839.56
At the end of 1 year, the investment is worth
$102,839.56.
2๏1
๏ฝ 100, 000 ๏จ1 ๏ซ 0.014๏ฉ
2
๏ฝ 100, 000 ๏จ1.014๏ฉ
32. a) A ๏ฝ P ๏จ1 ๏ซ i ๏ฉ
t
2
A ๏ฝ 300, 000 ๏จ1 ๏ซ 0.022๏ฉ
1
A ๏ฝ 100, 000 ๏จ1.028196๏ฉ
๏ฝ 102,819.60
At the end of 1 year, the investment is worth
$102,819.60.
nt
๏ฆ 0.028 ๏ถ
A ๏ฝ 100, 000 ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
๏ฝ 100, 000 ๏จ1 ๏ซ 0.07 ๏ฉ
๏ฆ i๏ถ
b) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
nt
2๏1
๏ฝ 306, 636.30
At the end of 1 year, the investment is worth
$306,636.30.
4
4
๏ฆ i๏ถ
c) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
๏ฝ 102,829.537
๏ป 102,829.54
At the end of 1 year, the investment is worth
$102,829.54.
nt
๏ฆ 0.028 ๏ถ
A ๏ฝ 100, 000 ๏ง1 ๏ซ
๏ท
๏จ
365 ๏ธ
๏ฝ 300, 000 ๏จ1.022๏ฉ ๏ฝ 306, 600.00
At the end of 1 year, the investment is worth
$306,600.00.
๏ฆ 0.022 ๏ถ
A ๏ฝ 300, 000 ๏ง1 ๏ซ
๏ท
๏จ
2 ๏ธ
4๏1
๏ฝ 100, 000 ๏จ1.0282953744๏ฉ
๏ฆ i๏ถ
d) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
8760๏1
๏ฝ 102,839.563811
๏ฆ 0.028 ๏ถ
A ๏ฝ 100, 000 ๏ง1 ๏ซ
๏ท
๏จ
2 ๏ธ
๏ฝ 100, 000 ๏จ1.07 ๏ฉ
nt
8760
๏ฝ 102,800
At the end of 1 year, the investment is worth
$102,800.
๏ฆ i๏ถ
c) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
๏ฆ i๏ถ
A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
๏ฆ 0.028 ๏ถ
A ๏ฝ 100, 000 ๏ง1 ๏ซ
๏จ 8760 ๏ท๏ธ
A ๏ฝ 100, 000 ๏จ1 ๏ซ 0.028๏ฉ
๏ฆ i๏ถ
b) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
e) There are 24 ๏ 365 ๏ฝ 8760 hours in one year.
nt
๏ฆ 0.022 ๏ถ
A ๏ฝ 300, 000 ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
๏ฝ 300, 000 ๏จ1.0055๏ฉ
4๏1
4
๏ป 306, 654.65
At the end of 1 year, the investment is worth
$306,654.65.
365๏1
๏ฝ 100, 000 ๏จ1 ๏ซ 0.00076712329๏ฉ
365
๏ฝ 100, 000 ๏จ1.00076712329๏ฉ
365
๏ฝ 100, 000 ๏จ1.02839458002๏ฉ
๏ฝ 102,839.458002
๏ป 102,839.46
At the end of 1 year, the investment is worth
$102,839.46.
๏ฆ i๏ถ
d) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
nt
๏ฆ 0.022 ๏ถ
A ๏ฝ 300, 000 ๏ง1 ๏ซ
๏ท
๏จ
365 ๏ธ
365๏1
๏ฝ 300, 000 ๏จ1.000060273973๏ฉ
365
๏ป 306, 672.93
At the end of 1 year, the investment is worth
$306,672.93.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.1
7
e) There are 24 ๏ 365 ๏ฝ 8760 hours in one year.
๏ฆ i๏ถ
A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
nt
๏ฝ 30, 000 ๏จ1.000109589๏ฉ
8760๏1
๏ฝ 30, 000 ๏จ1.127489438๏ฉ
๏ฝ 300, 000 ๏จ1.000002511416๏ฉ
8760
๏ป 306, 673.13
At the end of 1 year, the investment is worth
$306,673.13.
33. a) A ๏ฝ P ๏จ1 ๏ซ i ๏ฉ
A ๏ฝ 30, 000 ๏จ1 ๏ซ 0.04๏ฉ
3
๏ป 33,824.68
At the end of 3 year, the investment is worth
$33,824.68.
e) There are 24 ๏ 365 ๏ฝ 8760 hours in one year.
3
๏ฝ 33, 745.92
At the end of 3 year, the investment is worth
$33,745.92.
nt
๏ฆ 0.04 ๏ถ
A ๏ฝ 30, 000 ๏ง1 ๏ซ
๏ท
๏จ
2 ๏ธ
2๏3
๏ฝ 30, 000 ๏จ1.000004566210046๏ฉ
26,280
๏ฝ 30, 000 ๏จ1.127496541๏ฉ
๏ฝ 33,824.89624
๏ป 33,824.90
At the end of 3 year, the investment is worth
$33,824.90.
t
6
A ๏ฝ 1000 ๏จ1 ๏ซ 0.05๏ฉ
4
๏ฝ 30, 000 ๏จ1.1262๏ฉ
๏ฝ 33, 784.87
At the end of 3 year, the investment is worth
$33,784.87.
nt
๏ฝ 1215.51
At the end of 4 year, the investment is worth
$1512.51.
๏ฆ i๏ถ
b) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
nt
๏ฆ 0.05 ๏ถ
A ๏ฝ 1000 ๏ง1 ๏ซ
๏ท
๏จ
2 ๏ธ
4๏3
2๏ 4
๏ฝ 1000 ๏จ1.050625๏ฉ
8
๏ฝ 30, 000 ๏จ1.01๏ฉ
12
๏ฝ 30, 000 ๏จ1.12682503๏ฉ
๏ฝ 33,804.7509
๏ป 33,804.75
At the end of 3 year, the investment is worth
$33,804.75.
nt
8760๏3
34. a) A ๏ฝ P ๏จ1 ๏ซ i ๏ฉ
๏ฝ 30, 000 ๏จ1.02๏ฉ
๏ฆ 0.04 ๏ถ
A ๏ฝ 30, 000 ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
nt
๏ฆ 0.04 ๏ถ
A ๏ฝ 30, 000 ๏ง1 ๏ซ
๏จ 8760 ๏ท๏ธ
๏ฝ 30, 000 ๏จ1.04๏ฉ
๏ฆ i๏ถ
d) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
๏ฝ 33,824.68315
๏ฆ i๏ถ
A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
t
๏ฆ i๏ถ
c) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
365๏3
1095
๏ฆ 0.022 ๏ถ
A ๏ฝ 300, 000 ๏ง1 ๏ซ
๏จ 8760 ๏ท๏ธ
๏ฆ i๏ถ
b) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
๏ฆ 0.04 ๏ถ
A ๏ฝ 30, 000 ๏ง1 ๏ซ
๏จ 365 ๏ท๏ธ
๏ป 1218.40
At the end of 4 year, the investment is worth
$1218.40.
๏ฆ i๏ถ
c) A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
nt
๏ฆ 0.05 ๏ถ
A ๏ฝ 1000 ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
4๏ 4
๏ฝ 1000 ๏จ1.0125๏ฉ
16
๏ป 1219.89
At the end of 4 year, the investment is worth
$1219.89.
Copyright ยฉ 2020 Pearson Education, Inc.
8
d)
Chapter R Functions, Graphs, and Models
๏ฆ i๏ถ
A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
๏ฉ ๏จ1 ๏ซ r ๏ฉn ๏ญ 1๏น
37. W ๏ฝ P ๏ช
๏บ
r
๏ช๏ซ
๏บ๏ป
We substitute 3000 for P, 0.0305
๏จ3.05% ๏ฝ 0.0305๏ฉ for r, and 18 for n.
nt
๏ฆ 0.05 ๏ถ
A ๏ฝ 1000 ๏ง1 ๏ซ
๏จ 365 ๏ท๏ธ
365๏4
๏ฝ 1000 ๏จ1.000136986๏ฉ
1460
e)
๏ป 1221.39
At the end of 4 year, the investment is worth
$1221.39.
There are 24 ๏ 365 ๏ฝ 8760 hours in one year.
๏ฆ i๏ถ
A ๏ฝ P ๏ง1 ๏ซ ๏ท
๏จ n๏ธ
nt
๏ฆ 0.05 ๏ถ
A ๏ฝ 1000 ๏ง1 ๏ซ
๏จ 8760 ๏ท๏ธ
8760๏4
๏ฝ 1000 ๏จ1.000005708๏ฉ
35,040
๏ป 1221.40
At the end of 4 year, the investment is worth
$1221.40.
35. Using the formula:
n
๏ฉ r ๏ฆ
r ๏ถ ๏น
๏ช ๏ง1 ๏ซ ๏ท ๏บ
12 ๏จ 12 ๏ธ ๏บ
M ๏ฝ P๏ช
n
๏ช๏ฆ
๏บ
r ๏ถ
๏ช ๏ง1 ๏ซ ๏ท ๏ญ 1๏บ
๏ซ ๏จ 12 ๏ธ
๏ป
We substitute 18,000 for P , 0.046
๏จ4.6% ๏ฝ 0.046๏ฉ for r, and 36 ๏จ3 ๏12 ๏ฝ 36๏ฉ for n.
Then we use a calculator to perform the
computation.
๏ฉ 0.046 ๏ฆ 0.046 ๏ถ 36 ๏น
๏ช
๏ง1 ๏ซ
๏ท ๏บ
12 ๏จ
12 ๏ธ ๏บ
๏ช
M ๏ฝ 18, 000
๏ช ๏ฆ 0.046 ๏ถ 36
๏บ
๏ญ1 ๏บ
๏ช ๏ง1 ๏ซ
๏ท
12 ๏ธ
๏ซ ๏จ
๏ป
๏ป 536.25
The monthly payment on the loan will be
approximately $536.25.
36.
30 years ๏ฝ 30 ๏12 ๏ฝ 360 months
P ๏ฝ 100, 000; r ๏ฝ 0.024
๏ฉ 0.024 ๏ฆ 0.024 ๏ถ 360 ๏น
๏ช
๏ง1 ๏ซ
๏ท ๏บ
12 ๏จ
12 ๏ธ ๏บ
M ๏ฝ 100, 000 ๏ช
๏ช ๏ฆ 0.024 ๏ถ 360
๏บ
๏ญ1 ๏บ
๏ช ๏ง1 ๏ซ
๏ท
12 ๏ธ
๏ซ ๏จ
๏ป
๏ป 389.94
The monthly payment on the loan will be
approximately $389.94.
๏ฉ ๏จ1 ๏ซ 0.0305๏ฉ18 ๏ญ 1๏น
๏บ
W ๏ฝ 3000 ๏ช
0.0305
๏ซ๏ช
๏ป๏บ
๏ป 70,561.01
Rounded to the nearest cent, the annuity will be
worth $70,561.01 after 18 years.
38. We substitute 50,000 for W, 0.0725
๏ฆ 1
๏ถ
๏ง๏จ 4 % ๏ฝ 0.0425 ๏ท๏ธ in for r, and 20 for n. Then
4
we proceed to solve for P.
๏ฉ ๏จ1 ๏ซ 0.0425๏ฉ20 ๏ญ 1๏น
50, 000 ๏ฝ P ๏ช
๏บ
0.0425
๏ช๏ซ
๏บ๏ป
50, 000
๏ฝP
๏ฉ ๏จ1 ๏ซ 0.0425๏ฉ20 ๏ญ 1๏น
๏ช
๏บ
0.0425
๏ซ๏ช
๏ป๏บ
1635.99 ๏ป P
You will need to invest $1635.99 annually to
reach a goal of $50,000 after 20 years.
39. a) Locate 5.8 on the vertical axis and then
think of horizontal lines extending across the
graph from this point. The years for which
the graph lies above this line are the years
for which the unemployment rate was at or
above 5.8%. This time period was 2008 โ
2014.
b) Locate 7 on the vertical axis and then
think of a horizontal line extending across
the graph from this point. The years for
which the graph lies below this line are the
years for which unemployment was below
7%. Those time periods are 2006 โ 2008
and 2014 โ 2016.
c) Locate the highest point on the graph and
extend a line vertically to the horizontal
axis. The year which the unemployment rate
was the highest was 2010 at 9.6%.
d) Locate the lowest point on the graph and
extend a line vertically to the horizontal
axis. In this case there are two points that are
exactly at 4.6%. The years when the
unemployment rate was the lowest are 2006
and 2007.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.1
9
We substitute 0.053 ๏จ5.3% ๏ฝ 0.053๏ฉ for r, and
40. Answers will vary.
๏ฉ ๏จ1 ๏ซ r ๏ฉn ๏ญ 1๏น
41. a) Using the formula W ๏ฝ P ๏ช
๏บ we
r
๏ช๏ซ
๏บ๏ป
substitute 1200 for P, 0.04 ๏จ 4% ๏ฝ 0.04๏ฉ for r
and 35 for n.
๏ฉ ๏จ1 ๏ซ 0.04๏ฉ35 ๏ญ 1๏น
๏บ
W ๏ฝ 1200 ๏ช
0.04
๏ช๏ซ
๏บ๏ป
๏ป 88, 382.67
Sally will have approximately $88,382.67
in her account when she retires.
b) Sally invested $1200 per year for 35 years.
Therefore, the total amount of her original
payments is: $1200๏ง35 ๏ฝ $42, 000 . Since
the total amount in the account was
$88,382.67, the interest earned over the 35
years is:
$88,382.67 ๏ญ $42, 000 ๏ฝ $46,382.67
Therefore, $42,000 was the total amount of
Sallyโs payments and $46,382.67 was the
total amount of her interest.
42. a) Using the formula:
n
๏ฉ r ๏ฆ
r ๏ถ ๏น
1
๏ซ
๏ช ๏ง
๏ท ๏บ
12 ๏จ 12 ๏ธ ๏บ
M ๏ฝ P๏ช
n
๏ช๏ฆ
๏บ
r ๏ถ
๏ช ๏ง1 ๏ซ ๏ท ๏ญ 1๏บ
๏ซ ๏จ 12 ๏ธ
๏ป
We substitute 206,780.16 for P , 0.04
๏จ4% ๏ฝ 0.04๏ฉ for r, and 15 ๏จ80 ๏ญ 65 ๏ฝ 15๏ฉ for
n. Then we use a calculator to perform the
computation.
๏ฉ 0.04 ๏ฆ 0.04 ๏ถ180 ๏น
๏ช
๏ง1 ๏ซ
๏ท ๏บ
12 ๏จ
12 ๏ธ ๏บ
M ๏ฝ 206, 780.16 ๏ช
๏ช ๏ฆ 0.04 ๏ถ180
๏บ
๏ญ1 ๏บ
๏ช ๏ง1 ๏ซ
๏ท
12 ๏ธ
๏ซ ๏จ
๏ป
๏ป 653.76
Sally should take a monthly payment of
$653.76.
b) Sally received 180 payments of $653.76,
therefore she received a total of
$653.76๏ง180 ๏ฝ $117, 676.80 during the 15
years. Of that 42,000 was what she
originally contributed, leaving $75,676.80 in
interest.
43. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
12 (monthly) for n. Then we use a calculator to
perform the computation.
12
๏ฆ 0.053 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท ๏ญ1
๏จ
12 ๏ธ
๏ป 0.0543 ๏ฝ 5.43%
The annual yield of 5.3% compounded monthly
would be approximately 5.43%.
44. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.041 ๏จ 4.1% ๏ฝ 0.041๏ฉ for r, and 4
(quarterly) for n. Then we use a calculator to
perform the computation.
4
๏ฆ 0.041 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท ๏ญ1
๏จ
4 ๏ธ
๏ป 0.0416 ๏ฝ 4.16%
The annual yield of 4.1% compounded quarterly
would be approximately 4.16%.
45. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.0375 ๏จ3.75% ๏ฝ 0.0375๏ฉ for r,
and 52 (weekly) for n. Then we use a calculator
to perform the computation.
๏ฆ 0.0375 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท
๏จ
52 ๏ธ
52
๏ญ1
๏ป 0.0382 ๏ฝ 3.82%
The annual yield of 3.75% compounded weekly
would be approximately 3.82%.
46. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.04 ๏จ 4% ๏ฝ 0.04๏ฉ for r, and 360
(daily) for n. Then we use a calculator to
perform the computation.
๏ฆ 0.04 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏จ 360 ๏ท๏ธ
360
๏ญ1
๏ป 0.0408 ๏ฝ 4.08%
The annual yield of 4% compounded daily
would be approximately 4.08%.
Copyright ยฉ 2020 Pearson Education, Inc.
10
Chapter R Functions, Graphs, and Models
๏ฆ 0.0297 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท
๏จ
52 ๏ธ
47. a. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.025 ๏จ 2.5% ๏ฝ 0.025๏ฉ for r,
and 1 (annually) for n. Then we use a
calculator to perform the computation.
52
๏ญ1
๏ป 0.0301 ๏ฝ 3.01%
The annual yield for Foothill Bank will be
approximately 3.01%
b. Foothill Bank has a higher yield.
49. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
1
๏ฆ 0.025 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท ๏ญ1
๏จ
1 ๏ธ
๏ป 0.025 ๏ฝ 2.5%
The annual yield for Western Bank will be
2.5%.
Using the formula:
We substitute 0.022 ๏จ 2.2% ๏ฝ 0.022๏ฉ for Y, and
12 (monthly) for n. Then we use a calculator to
perform the computation.
12
r ๏ถ
๏ฆ
0.022 ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ 12 ๏ธ
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.0243 ๏จ2.43% ๏ฝ 0.0243๏ฉ for
r, and 12 (monthly) for n. Then we use a
calculator to perform the computation.
r ๏ป 0.02179 ๏ฝ 2.179%
Mesalands would need to pay at least 2.179% to
exceed an annual yield of 2.2%.
50. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
12
๏ฆ 0.0243 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท ๏ญ1
๏จ
12 ๏ธ
๏ป 0.0246 ๏ฝ 2.46%
The annual yield for Commonwealth
Savings will be approximately 2.46%
b. Western Bank has a higher yield.
We substitute 0.0375 ๏จ3.75% ๏ฝ 0.0375๏ฉ for Y,
and 4 (quarterly) for n. Then we use a calculator
to perform the computation.
4
๏ฆ r๏ถ
0.0375 ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ 4๏ธ
48. a. Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.03 ๏จ3% ๏ฝ 0.03๏ฉ for r, and 1
(annually) for n. Then we use a calculator to
perform the computation.
r ๏ป 0.03698 ๏ฝ 3.698%
Shea Savings would need to pay at least 3.698%
to exceed an annual yield of 3.75%.
51. Graph y ๏ฝ x3 ๏ญ x 2
The resulting graph is:
1
๏ฆ 0.03 ๏ถ
Y ๏ฝ ๏ง1 ๏ซ
๏ท ๏ญ1
๏จ
1 ๏ธ
๏ป 0.03 ๏ฝ 3%
The annual yield for Sierra Savings will be
3%.
Using the formula:
n
๏ฆ r๏ถ
Y ๏ฝ ๏ง1 ๏ซ ๏ท ๏ญ 1
๏จ n๏ธ
We substitute 0.0297 ๏จ2.97% ๏ฝ 0.0297๏ฉ for
r, and 52 (weekly) for n. Then we use a
calculator to perform the computation.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.1
11
52. Graph y ๏ฝ 2 x
The resulting graph is
Copyright ยฉ 2020 Pearson Education, Inc.
12
Chapter R Functions, Graphs, and Models
Exercise Set R.2
13. The correspondence is a function because every
book has a unique Library of Congress number.
1.
14. The correspondence is a function because every
book has a unique ISBN.
The correspondence is a function because each
member of the domain corresponds to only one
member of the range.
2.
The correspondence is a function because each
member of the domain corresponds to only one
member of the range.
3.
The correspondence is a function because each
member of the domain corresponds to only one
member of the range.
4.
The correspondence is not a function because
one member of the domain, 6, corresponds to
two members of the range, โ 6 and โ 7.
5.
The correspondence is a function because each
member of the domain corresponds to only one
member of the range, even though two members
of the domain, Quarter Pounder with Cheese ยฎ
and Big Nโ Tasty with Cheese ยฎ correspond to
$3.20.
6.
7.
The correspondence is a function because each
member of the domain corresponds to only one
member of the range.
The correspondence is a function because each
number doubled is exactly one number.
8.
The correspondence is a function because each
number that is two less than the number is a
number.
9.
The correspondence is a function because each
square root of a number is exactly one positive
number.
10. The correspondence is a function because each
cubed root of a number is exactly one number.
11. The correspondence is not a function because
there may be more than one number less than or
equal to a positive number.
15. This correspondence is a function, because all
people have a specific birthday.
16. This correspondence is a function, because all
people have a specific weight when they step on
a scale.
17. The correspondence is not a function because
more than one person is born on any specific
date.
18. The correspondence is not a function because
more multiple people may have the same
weight.
19. This correspondence is a function, because a
rectangles area is determined by the specific
length and width.
20. This correspondence is a function, because a
rectangles perimeter is determined by the
specific length and width.
21. The correspondence is not a function because
there may be more than one length and width to
obtain the given area.
22. The correspondence is not a function because
there may be more than one length and width to
obtain the given perimeter.
23. a)
f ๏จ x๏ฉ ๏ฝ 4x ๏ญ 3
f ๏จ5.1๏ฉ ๏ฝ 4 ๏จ5.1๏ฉ ๏ญ 3 ๏ฝ 17.4
f ๏จ5.01๏ฉ ๏ฝ 4 ๏จ5.01๏ฉ ๏ญ 3 ๏ฝ 17.04
f ๏จ5.001๏ฉ ๏ฝ 4 ๏จ5.001๏ฉ ๏ญ 3 ๏ฝ 17.004
f ๏จ5๏ฉ ๏ฝ 4 ๏จ5๏ฉ ๏ญ 3 ๏ฝ 17
x
f ๏จ x๏ฉ
5.1
17.4
12. The correspondence is not a function because
there is more than one odd integer less than or
equal to the given number.
Copyright ยฉ 2020 Pearson Education, Inc.
5.01
17.04
5.001
17.004
5
17
Exercise Set R.2
13
b) f ๏จ x ๏ฉ ๏ฝ 4 x ๏ญ 3
26.
f ๏จ 4๏ฉ ๏ฝ 4 ๏จ 4๏ฉ ๏ญ 3 ๏ฝ 13
g ๏จ x๏ฉ ๏ฝ x2 ๏ซ 4
g ๏จ ๏ญ3๏ฉ ๏ฝ ๏จ๏ญ3๏ฉ ๏ซ 4 ๏ฝ 13
2
f ๏จ3๏ฉ ๏ฝ 4 ๏จ3๏ฉ ๏ญ 3 ๏ฝ 9
g ๏จ0 ๏ฉ ๏ฝ ๏จ0 ๏ฉ ๏ซ 4 ๏ฝ 4
2
f ๏จ ๏ญ2๏ฉ ๏ฝ 4 ๏จ ๏ญ2๏ฉ ๏ญ 3 ๏ฝ ๏ญ11
g ๏จ๏ญ1๏ฉ ๏ฝ ๏จ๏ญ1๏ฉ ๏ซ 4 ๏ฝ 5
2
f ๏จ k ๏ฉ ๏ฝ 4 ๏จ k ๏ฉ ๏ญ 3 ๏ฝ 4k ๏ญ 3
g ๏จ7 ๏ฉ ๏ฝ ๏จ7 ๏ฉ ๏ซ 4 ๏ฝ 53
2
f ๏จ x ๏ซ h ๏ฉ ๏ฝ 4 ๏จ x ๏ซ h ๏ฉ ๏ญ 3 ๏ฝ 4 x ๏ซ 4h ๏ญ 3
24. a)
g ๏จ a ๏ซ h ๏ฉ ๏ฝ ๏จa ๏ซ h ๏ฉ ๏ซ 4 ๏ฝ a 2 ๏ซ 2ah ๏ซ h 2 ๏ซ 4
2
f ๏จ x ๏ฉ ๏ฝ 3x ๏ซ 2
x
f ๏จ x๏ฉ
4.1
14.3
2
4.01
14.03
4.001
14.003
4
14
b) f ๏จ x ๏ฉ ๏ฝ 3x ๏ซ 2
f ๏จ5๏ฉ ๏ฝ 3 ๏จ5๏ฉ ๏ซ 2 ๏ฝ 17
27.
f ๏จ๏ญ1๏ฉ ๏ฝ 3 ๏จ ๏ญ1๏ฉ ๏ซ 2 ๏ฝ ๏ญ1
f ๏จ k ๏ฉ ๏ฝ 3 ๏จ k ๏ฉ ๏ซ 2 ๏ฝ 3k ๏ซ 2
f ๏จ x ๏ซ h ๏ฉ ๏ฝ 3 ๏จ x ๏ซ h ๏ฉ ๏ซ 2 ๏ฝ 3x ๏ซ 3h ๏ซ 2
25.
f ๏จ x๏ฉ ๏ฝ
f ๏จ 4๏ฉ ๏ฝ
f ๏จ0 ๏ฉ ๏ฝ
g ๏จ x๏ฉ ๏ฝ x ๏ญ 3
2
g ๏จ ๏ญ1๏ฉ ๏ฝ ๏จ ๏ญ1๏ฉ ๏ญ 3 ๏ฝ 1 ๏ญ 3 ๏ฝ ๏ญ2
2
f ๏จa ๏ฉ ๏ฝ
g ๏จ0๏ฉ ๏ฝ ๏จ0๏ฉ ๏ญ 3 ๏ฝ 0 ๏ญ 3 ๏ฝ ๏ญ3
2
g ๏จ1๏ฉ ๏ฝ ๏จ1๏ฉ ๏ญ 3 ๏ฝ 1 ๏ญ 3 ๏ฝ ๏ญ2
๏จ
g ๏จ a ๏ซ h ๏ฉ ๏ญ g ๏จa ๏ฉ a ๏ซ 2ah ๏ซ h ๏ซ 4 ๏ญ a ๏ซ 4
๏ฝ
h
h
2
2ah ๏ซ h
๏ฝ
h
๏ฝ 2a ๏ซ h
2
2
1
๏จ x ๏ซ 3๏ฉ2
1
2
๏ฝ
2
๏ฝ
๏จ๏จ4๏ฉ ๏ซ 3๏ฉ
1
๏จ๏จ0๏ฉ ๏ซ 3๏ฉ
1
๏จ๏จa ๏ฉ ๏ซ 3๏ฉ
๏ฝ
2
1
2
๏ฝ
1
49
2
๏ฝ
1
9
๏จ7 ๏ฉ
1
๏จ3๏ฉ
1
๏จa ๏ซ 3๏ฉ2
2
f ๏จ x ๏ซ h๏ฉ ๏ฝ
g ๏จ5๏ฉ ๏ฝ ๏จ5๏ฉ ๏ญ 3 ๏ฝ 25 ๏ญ 3 ๏ฝ 22
2
g ๏จa ๏ซ h ๏ฉ ๏ฝ ๏จa ๏ซ h ๏ฉ ๏ญ 3 ๏ฝ a ๏ซ 2ah ๏ซ h ๏ญ 3
2
2
2
2
๏ฉ 2 ๏น
g ๏จ a ๏ซ h ๏ฉ ๏ญ g ๏จ a ๏ฉ ๏จ a ๏ซ h ๏ฉ ๏ญ 3 ๏ญ ๏ซ๏จ a ๏ฉ ๏ญ 3๏ป
๏ฝ
h
h
2
a ๏ซ 2ah ๏ซ h 2 ๏ญ 3 ๏ญ ๏ฉ๏ซ a 2 ๏ญ 3๏น๏ป
๏ฝ
h
2
2ah ๏ซ h
๏ฝ
h
h ๏จ 2a ๏ซ h ๏ฉ
๏ฝ
h
๏ฝ 2a ๏ซ h
1
๏ฝ
๏จ๏จ x ๏ซ h ๏ฉ ๏ซ 3๏ฉ
2
1
๏จ x ๏ซ h ๏ซ 3๏ฉ2
f ๏จ x ๏ซ h๏ฉ ๏ญ f ๏จ x๏ฉ
h
1
1
๏ญ
2
๏ซ
๏ซ
๏ซ
x
h
x
3
๏จ
๏ฉ ๏จ 3๏ฉ2
๏ฝ
h
๏ฝ
๏จ x ๏ซ 3๏ฉ2
๏จ x ๏ซ h ๏ซ 3๏ฉ2
๏ญ
๏จ x ๏ซ h ๏ซ 3๏ฉ2 ๏จ x ๏ซ 3๏ฉ2 ๏จ x ๏ซ h ๏ซ 3๏ฉ2 ๏จ x ๏ซ 3๏ฉ2
๏จ
2
๏ฝ
๏ฝ
๏ฝ
๏ฝ
h
2
x ๏ซ 6 x ๏ซ 9 ๏ญ x ๏ซ 2hx ๏ซ 6 x ๏ซ h 2 ๏ซ 6h ๏ซ 9
h ๏จ x ๏ซ h ๏ซ 3๏ฉ ๏จ x ๏ซ 3๏ฉ
2
๏ญ2hx ๏ญ h 2 ๏ญ 6h
h ๏จ x ๏ซ h ๏ซ 3๏ฉ ๏จ x ๏ซ 3๏ฉ
2
2
h ๏จ๏ญ2 x ๏ญ h ๏ญ 6๏ฉ
h ๏จ x ๏ซ h ๏ซ 3๏ฉ ๏จ x ๏ซ 3๏ฉ
2
๏ญ2 x ๏ญ h ๏ญ 6
๏จ x ๏ซ h ๏ซ 3๏ฉ2 ๏จ x ๏ซ 3๏ฉ2
Copyright ยฉ 2020 Pearson Education, Inc.
2
,
h๏น0
2
๏ฉ
๏ฉ
14
28.
Chapter R Functions, Graphs, and Models
f ๏จ x๏ฉ ๏ฝ
f ๏จ3๏ฉ ๏ฝ
33. Graph f ๏จ x ๏ฉ ๏ฝ 2 x ๏ญ 5 .
1
๏จ x ๏ญ 5๏ฉ
2
1
๏จ3 ๏ญ 5๏ฉ
f ๏จ๏ญ1๏ฉ ๏ฝ
f ๏จk ๏ฉ ๏ฝ
2
๏ฝ
1
๏จ๏ญ2๏ฉ
1
๏จ ๏ญ1 ๏ญ 5๏ฉ
2
2
๏ฝ
๏ฝ
1
๏จ๏ญ6๏ฉ
2
First, we choose some values for x and compute
the values for f ๏จ x ๏ฉ , in order to form the
1
4
๏ฝ
ordered pairs that we will plot on the graph.
f ๏จ ๏ญ1๏ฉ ๏ฝ 2 ๏จ ๏ญ1๏ฉ ๏ญ 5 ๏ฝ ๏ญ7
1
36
f ๏จ0๏ฉ ๏ฝ 2 ๏จ0๏ฉ ๏ญ 5 ๏ฝ ๏ญ5
f ๏จ1๏ฉ ๏ฝ 2 ๏จ1๏ฉ ๏ญ 5 ๏ฝ ๏ญ3
1
f ๏จ 2๏ฉ ๏ฝ 2 ๏จ 2๏ฉ ๏ญ 5 ๏ฝ ๏ญ1
๏จ k ๏ญ 5 ๏ฉ2
f ๏จ x ๏ซ h๏ฉ ๏ฝ
1
๏จ๏จ x ๏ซ h ๏ฉ ๏ญ 5 ๏ฉ
2
๏ฝ
29. a. Graph f ๏จ x ๏ฉ ๏ฝ 4 x ๏ซ 2 .
b.
1
๏จ x ๏ซ h ๏ญ 5๏ฉ2
x
f ๏จ x๏ฉ
๏ญ1
๏ญ7
0
๏ญ5
1
๏ญ3
๏จ x, f ๏จ x ๏ฉ๏ฉ
๏จ๏ญ1, ๏ญ7๏ฉ
๏จ0, ๏ญ5๏ฉ
๏จ1, ๏ญ3๏ฉ
๏จ2, ๏ญ1๏ฉ
2
๏ญ1
Next we plot the input โ output pairs from the
table and, in this case, draw the line to complete
the graph.
30. a. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ3x ๏ญ 4 .
b.
34. Graph f ๏จ x ๏ฉ ๏ฝ 3x ๏ญ 1 .
31. a. Graph h ๏จ x ๏ฉ ๏ฝ x 2 ๏ซ x .
b.
35. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ4 x .
First, we choose some values for x and compute
the values for g ๏จ x ๏ฉ , in order to form the
32. a. Graph k ๏จ x ๏ฉ ๏ฝ x 2 ๏ญ 3 x .
b.
ordered pairs that we will plot on the graph.
g ๏จ ๏ญ1๏ฉ ๏ฝ ๏ญ4 ๏จ ๏ญ1๏ฉ ๏ฝ 4
g ๏จ0๏ฉ ๏ฝ ๏ญ4 ๏จ0๏ฉ ๏ฝ 0
g ๏จ1๏ฉ ๏ฝ ๏ญ4 ๏จ1๏ฉ ๏ฝ ๏ญ4 .
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.2
15
x
g ๏จ x๏ฉ
๏ญ1
4
0
0
๏จ x, g ๏จ x ๏ฉ๏ฉ
๏จ๏ญ1, 4๏ฉ
๏จ0, 0๏ฉ
๏จ0, ๏ญ 4๏ฉ
๏ญ4
1
Next we plot the input โ output pairs from the
table and, in this case, draw the line to complete
the graph.
38. Graph f ๏จ x ๏ฉ ๏ฝ x 2 ๏ซ 4 .
36. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ2 x .
39. Graph f ๏จ x ๏ฉ ๏ฝ 6 ๏ญ x 2 .
First, we choose some values for x and compute
the values for f ๏จ x ๏ฉ , in order to form the
ordered pairs that we will plot on the graph.
f ๏จ ๏ญ2๏ฉ ๏ฝ 6 ๏ญ ๏จ๏ญ2๏ฉ ๏ฝ 2
2
f ๏จ ๏ญ1๏ฉ ๏ฝ 6 ๏ญ ๏จ๏ญ1๏ฉ ๏ฝ 5
2
37. Graph f ๏จ x ๏ฉ ๏ฝ x 2 ๏ญ 2 .
f ๏จ0๏ฉ ๏ฝ 6 ๏ญ ๏จ0 ๏ฉ ๏ฝ 6
2
First, we choose some values for x and compute
the values for f ๏จ x ๏ฉ , in order to form the
ordered pairs that we will plot on the graph.
Choosing some values for x and evaluating the
function, we have:
f ๏จ ๏ญ2๏ฉ ๏ฝ ๏จ ๏ญ2๏ฉ ๏ญ 2 ๏ฝ 2
2
f ๏จ ๏ญ1๏ฉ ๏ฝ ๏จ ๏ญ1๏ฉ ๏ญ 2 ๏ฝ ๏ญ1
2
f ๏จ0๏ฉ ๏ฝ ๏จ0๏ฉ ๏ญ 2 ๏ฝ ๏ญ2
2
f ๏จ1๏ฉ ๏ฝ ๏จ1๏ฉ ๏ญ 2 ๏ฝ ๏ญ1
2
f ๏จ 2๏ฉ ๏ฝ ๏จ 2๏ฉ ๏ญ 2 ๏ฝ 2
2
x
f ๏จ x๏ฉ
๏ญ2
2
๏ญ1
๏ญ1
0
๏ญ2
1
๏ญ1
๏จ x, f ๏จ x ๏ฉ๏ฉ
๏จ๏ญ2, 2๏ฉ
๏จ๏ญ1, ๏ญ1๏ฉ
๏จ0, ๏ญ2๏ฉ
๏จ1, ๏ญ1๏ฉ
๏จ2, 2๏ฉ
f ๏จ1๏ฉ ๏ฝ 6 ๏ญ ๏จ1๏ฉ ๏ฝ 5
2
f ๏จ 2๏ฉ ๏ฝ 6 ๏ญ ๏จ 2 ๏ฉ ๏ฝ 2
2
x
f ๏จ x๏ฉ
๏ญ2
2
๏ญ1
5
0
6
1
5
๏จ x, f ๏จ x ๏ฉ๏ฉ
๏จ๏ญ2, 2๏ฉ
๏จ๏ญ1,5๏ฉ
๏จ0, 6๏ฉ
๏จ1,5๏ฉ
๏จ2, 2๏ฉ
2
2
Next we plot the input โ output pairs from the
table and, in this case, draw the curve to
complete the graph.
2
2
Next we plot the input โ output pairs from the
table and, in this case, draw the curve to
complete the graph.
Copyright ยฉ 2020 Pearson Education, Inc.
16
Chapter R Functions, Graphs, and Models
40. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ x 2 ๏ซ 1 .
42. Graph g ๏จ x ๏ฉ ๏ฝ
41. Graph g ๏จ x ๏ฉ ๏ฝ x3 .
First, we choose some values for x and compute
the values for g ๏จ x ๏ฉ , in order to form the
ordered pairs that we will plot on the graph.
g ๏จ๏ญ2๏ฉ ๏ฝ ๏จ ๏ญ2๏ฉ ๏ฝ ๏ญ8
3
g ๏จ ๏ญ1๏ฉ ๏ฝ ๏จ ๏ญ1๏ฉ ๏ฝ ๏ญ1
3
g ๏จ 0 ๏ฉ ๏ฝ ๏จ0 ๏ฉ ๏ฝ 0
3
1 3
x .
2
43. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
44. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
45. The graph is not that of a function. A vertical
line can intersect the graph more than once.
g ๏จ1๏ฉ ๏ฝ ๏จ1๏ฉ ๏ฝ 1
3
g ๏จ 2๏ฉ ๏ฝ ๏จ 2๏ฉ ๏ฝ 8
3
x
f ๏จ x๏ฉ
๏ญ2
๏ญ8
๏ญ1
๏ญ1
0
0
1
1
2
8
๏จ x, f ๏จ x ๏ฉ๏ฉ
๏จ๏ญ2, ๏ญ8๏ฉ
๏จ๏ญ1, ๏ญ1๏ฉ
๏จ0, 0๏ฉ
๏จ1,1๏ฉ
๏จ2,8๏ฉ
Next we plot the input โ output pairs from the
table above and, in this case, draw the curve to
complete the graph.
46. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
47. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
48. The graph is not that of a function. A vertical
line can intersect the graph more than once.
49. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
50. The graph is not that of a function. A vertical
line can intersect the graph more than once.
51. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
52. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
53. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
54. The graph is a function, it is impossible to draw
a vertical line that intersects the graph more
than once.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.2
17
b) The graph is not that of a function. A
vertical line can intersect the graph more
than once.
55. The graph is not that of a function. A vertical
line can intersect the graph more than once.
56. The graph is not that of a function. A vertical
line can intersect the graph more than once.
59.
f ๏จ x ๏ซ h๏ฉ ๏ญ f ๏จ x ๏ฉ
h
57. Graph x ๏ฝ y 2 ๏ญ 2 .
a) First, we choose some values for y (since x
is expressed in terms of y) and compute the
values for x , in order to form the ordered
pairs that we will plot on the graph.
๏ฝ
2
2
For y ๏ฝ 1; x ๏ฝ ๏จ1๏ฉ ๏ญ 2 ๏ฝ ๏ญ1
2
For y ๏ฝ 2; x ๏ฝ ๏จ2๏ฉ ๏ญ 2 ๏ฝ 2
2
๏ญ2
๏ญ1
๏ญ1
๏ญ2
0
๏ญ1
1
2
2
Next we plot the input โ output pairs from
the table and, in this case, draw the curve to
complete the graph.
2
h
2
2 xh ๏ซ h ๏ญ 3h
combining
like terms
h
h ๏จ 2 x ๏ซ h ๏ญ 3๏ฉ
Factoring
๏ฝ
h
๏ฝ 2 x ๏ซ h ๏ญ 3, h ๏น 0
For y ๏ฝ 0; x ๏ฝ ๏จ0๏ฉ ๏ญ 2 ๏ฝ ๏ญ2
2
2
๏ฝ
2
๏จ x, y ๏ฉ
๏จ2, ๏ญ2๏ฉ
๏จ๏ญ1, ๏ญ1๏ฉ
๏จ๏ญ2, 0๏ฉ
๏จ๏ญ1,1๏ฉ
๏จ2, 2๏ฉ
h
x ๏ซ 2 xh ๏ซ h ๏ญ 3 x ๏ญ 3h ๏ญ ๏ฉ๏ซ x ๏ญ 3 x ๏น๏ป
๏ฝ
For y ๏ฝ ๏ญ1; x ๏ฝ ๏จ ๏ญ1๏ฉ ๏ญ 2 ๏ฝ ๏ญ1
y
๏จ x ๏ซ h๏ฉ2 ๏ญ 3 ๏จ x ๏ซ h ๏ฉ ๏ญ ๏ฉ๏ซ x 2 ๏ญ 3x ๏น๏ป
2
For y ๏ฝ ๏ญ2; x ๏ฝ ๏จ ๏ญ2๏ฉ ๏ญ 2 ๏ฝ 2
x
f ๏จ x ๏ฉ ๏ฝ x2 ๏ญ 3x
60.
f ๏จ x๏ฉ ๏ฝ x2 ๏ซ 4x
f ๏จ x ๏ซ h๏ฉ ๏ญ f ๏จ x๏ฉ
h
๏ฝ
๏จ x ๏ซ h๏ฉ2 ๏ซ 4 ๏จ x ๏ซ h๏ฉ ๏ญ ๏ฉ๏ซ x 2 ๏ซ 4 x ๏น๏ป
h
x ๏ซ 2 xh ๏ซ h ๏ซ 4 x ๏ซ 4h ๏ญ ๏ฉ๏ซ x
2
๏ฝ
2
2
๏ซ 4 x ๏น๏ป
h
2 xh ๏ซ h 2 ๏ซ 4h
๏ฝ
h
h ๏จ2 x ๏ซ h ๏ซ 4๏ฉ
๏ฝ
h
๏ฝ 2 x ๏ซ h ๏ซ 4, h ๏น 0
61. To find f ๏จ ๏ญ1๏ฉ we need to locate which piece
b) The graph is not that of a function. A
vertical line can intersect the graph more
than once.
58. a) Graph x ๏ฝ y 2 ๏ญ 3 .
defines the function on the domain that contains
x ๏ฝ ๏ญ1 . When x ๏ฝ ๏ญ1 , the function is defined
by f ๏จ x ๏ฉ ๏ฝ ๏ญ2 x ๏ซ 1; for x ๏ผ 0 ; therefore,
f ๏จ ๏ญ1๏ฉ ๏ฝ ๏ญ2 ๏จ ๏ญ1๏ฉ ๏ซ 1 ๏ฝ 2 ๏ซ 1 ๏ฝ 3 .
To find f ๏จ1๏ฉ we need to locate which piece
defines the function on the domain that contains
x ๏ฝ 1 . When x ๏ฝ 1 , the function is defined by
f ๏จ x ๏ฉ ๏ฝ x 2 ๏ญ 3; for 0 ๏ผ x ๏ผ 4 ; therefore,
f ๏จ1๏ฉ ๏ฝ ๏จ1๏ฉ ๏ญ 3 ๏ฝ 1 ๏ญ 3 ๏ฝ ๏ญ2 .
2
Copyright ยฉ 2020 Pearson Education, Inc.
18
Chapter R Functions, Graphs, and Models
62. When x ๏ฝ ๏ญ3 , the function is defined by
f ๏จ x ๏ฉ ๏ฝ ๏ญ2 x ๏ซ 1; for x ๏ผ 0 ; therefore,
Note that for f ๏จ x ๏ฉ ๏ฝ ๏ญ1 .
f ๏จ ๏ญ3๏ฉ ๏ฝ ๏ญ2 ๏จ ๏ญ3๏ฉ ๏ซ 1 ๏ฝ 6 ๏ซ 1 ๏ฝ 7 .
When x ๏ฝ 3 , the function is defined by
f ๏จ x ๏ฉ ๏ฝ x 2 ๏ญ 3; for 0 ๏ผ x ๏ผ 4 ; therefore,
f ๏จ3๏ฉ ๏ฝ ๏จ3๏ฉ ๏ญ 3 ๏ฝ 9 ๏ญ 3 ๏ฝ 6 .
2
f ๏จ 0 ๏ฉ ๏ฝ ๏ญ1
f ๏จ1๏ฉ ๏ฝ ๏ญ1
f ๏จ 2๏ฉ ๏ฝ ๏ญ1
The solid dot indicates that ๏จ0, ๏ญ1๏ฉ is part of the
graph.
63. To find f ๏จ0๏ฉ we need to locate which piece
defines the function on the domain that contains
x ๏ฝ 0.
When x ๏ฝ 0 , the function is defined by
f ๏จ x ๏ฉ ๏ฝ 17; for x ๏ฝ 0 ; therefore,
f ๏จ0๏ฉ ๏ฝ 17 .
To find f ๏จ10๏ฉ we need to locate which piece
defines the function on the domain that contains
x ๏ฝ 10 . When x ๏ฝ 10 , the function is defined
1
by f ๏จ x ๏ฉ ๏ฝ x ๏ซ 1; for x ๏ณ 4 ; therefore,
2
1
f ๏จ10๏ฉ ๏ฝ ๏จ10๏ฉ ๏ซ 1 ๏ฝ 5 ๏ซ 1 ๏ฝ 6 .
2
64. When x ๏ฝ ๏ญ5 , the function is defined by
f ๏จ x ๏ฉ ๏ฝ ๏ญ2 x ๏ซ 1; for x ๏ผ 0 ; therefore,
f ๏จ ๏ญ5๏ฉ ๏ฝ ๏ญ2 ๏จ ๏ญ5๏ฉ ๏ซ 1 ๏ฝ 10 ๏ซ 1 ๏ฝ 11 .
When x ๏ฝ 5 , the function is defined by
1
f ๏จ x ๏ฉ ๏ฝ x ๏ซ 1; for x ๏ณ 4 ; therefore,
2
1
5
7
f ๏จ5๏ฉ ๏ฝ ๏จ5๏ฉ ๏ซ 1 ๏ฝ ๏ซ 1 ๏ฝ ๏ฝ 3.5 .
2
2
2
๏ฌ 1 for x ๏ผ 0
65. Graph f ๏จ x ๏ฉ ๏ฝ ๏ญ
.
๏ฎ ๏ญ1 for x ๏ณ 0
First, we graph f ๏จ x ๏ฉ ๏ฝ 1 for inputs less than 0.
We note for any x-value less than 0, the graph is
the horizontal line y ๏ฝ 1 . Note that for f ๏จ x ๏ฉ ๏ฝ 1
f ๏จ๏ญ2๏ฉ ๏ฝ 1
๏ฌ 2, for x ๏ฃ 3
66. Graph f ๏จ x ๏ฉ ๏ฝ ๏ญ
.
๏ฎ ๏ญ2, for x ๏พ 3
๏ฌ6, for x ๏ฝ ๏ญ2
67. Graph f ๏จ x ๏ฉ ๏ฝ ๏ญ 2
.
๏ฎ x , for x ๏น ๏ญ2
First, we graph f ๏จ x ๏ฉ ๏ฝ 6 for x ๏ฝ ๏ญ2 .
This graph consists of only one point, ๏จ๏ญ2, 6๏ฉ .
The solid dot indicates that ๏จ๏ญ2, 6๏ฉ is part of the
graph.
Next, we graph f ๏จ x ๏ฉ ๏ฝ x 2 for inputs x ๏น ๏ญ2 .
Note that for f ๏จ x ๏ฉ ๏ฝ x 2
f ๏จ ๏ญ3๏ฉ ๏ฝ ๏จ ๏ญ3๏ฉ ๏ฝ 9
2
f ๏จ ๏ญ1๏ฉ ๏ฝ ๏จ ๏ญ1๏ฉ ๏ฝ 1
2
f ๏จ 0๏ฉ ๏ฝ ๏จ0 ๏ฉ ๏ฝ 0
2
f ๏จ1๏ฉ ๏ฝ ๏จ1๏ฉ ๏ฝ 1
2
f ๏จ๏ญ1๏ฉ ๏ฝ 1
The solution is continued on the next page.
The open circle indicates that ๏จ0,1๏ฉ is not part of
f ๏จ 2๏ฉ ๏ฝ ๏จ 2 ๏ฉ ๏ฝ 4
2
the graph.
Next, we graph f ๏จ x ๏ฉ ๏ฝ ๏ญ1 for inputs greater
than or equal to 0. We note for any x-value less
than 0, the graph is the horizontal line y ๏ฝ ๏ญ1 .
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.2
19
x
f ๏จ x๏ฉ
๏ญ3
9
๏ญ1
1
0
0
1
1
๏จ x, f ๏จ x ๏ฉ๏ฉ
๏จ๏ญ3,9๏ฉ
๏จ๏ญ1,1๏ฉ
๏จ0, 0๏ฉ
๏จ1,1๏ฉ
๏จ2, 4๏ฉ
2
4
Since the input x ๏ฝ ๏ญ2 is not defined on this
part of the graph, the point ๏จ๏ญ2, 4๏ฉ is not part of
Next, we graph g ๏จ x ๏ฉ ๏ฝ x ๏ซ 2 for inputs x ๏พ 0 .
Creating the input โ output table, we have:
g ๏จ x๏ฉ
x
๏จ x, g ๏จ x ๏ฉ๏ฉ
๏จ1,3๏ฉ
๏จ2, 4๏ฉ
2
4
๏จ3,5๏ฉ
3
5
The open circle indicates that ๏จ0, 2๏ฉ is not part
1
3
of the graph.
the graph. The open circle indicates that
๏จ๏ญ2, 4๏ฉ is not part of the graph.
๏ฌ5, for x ๏ฝ 1
68. Graph f ๏จ x ๏ฉ ๏ฝ ๏ญ 3
.
๏ฎ x , for x ๏น 1
๏ฌ๏ญ x, for x ๏ผ 0
๏ฏ
for x ๏ฝ 0 .
69. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ4,
๏ฏ x ๏ซ 2, for x ๏พ 0
๏ฎ
First, we graph g ๏จ x ๏ฉ ๏ฝ ๏ญ x for inputs x ๏ผ 0 .
Creating the input โ output table, we have:
g ๏จ x๏ฉ
x
๏จ x, g ๏จ x ๏ฉ๏ฉ
๏จ๏ญ3,3๏ฉ
๏จ๏ญ2, 2๏ฉ
๏ญ2
2
๏จ๏ญ1,1๏ฉ
๏ญ1
1
The open circle indicates that ๏จ0, 0๏ฉ is not part
๏ญ3
3
of the graph.
Next, we graph g ๏จ x ๏ฉ ๏ฝ 4 for x ๏ฝ 0 . This part
of the graph consists of a single point. The solid
dot indicates that ๏จ0, 4๏ฉ is part of the graph.
๏ฌ2 x ๏ญ 3,
๏ฏ
70. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ5,
๏ฏ x ๏ญ 2,
๏ฎ
for x ๏ผ 1
for x ๏ฝ 1 .
๏ฌ 12 x ๏ญ 1,
๏ฏ
71. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ ๏ญ4,
๏ฏ x ๏ญ 3,
๏ฎ
for x ๏ผ 2
for x ๏พ 1
for x ๏ฝ 2 .
for x ๏พ 2
First, we graph g ๏จ x ๏ฉ ๏ฝ 12 x ๏ญ 1 for inputs x ๏ผ 2 .
Creating the input โ output table, we have:
x
g ๏จ x๏ฉ
๏ญ2
๏ญ2
0
๏ญ1
1
๏ญ 12
๏จ x, g ๏จ x ๏ฉ๏ฉ
๏จ๏ญ2, ๏ญ2๏ฉ
๏จ0, ๏ญ1๏ฉ
๏จ1, ๏ญ 12 ๏ฉ
The open circle indicates that ๏จ2, 0๏ฉ is not part
of the graph.
Next, we graph g ๏จ x ๏ฉ ๏ฝ ๏ญ 4 for x ๏ฝ 2 . This part
of the graph consists of a single point. The solid
dot indicates that ๏จ 2, ๏ญ 4๏ฉ is part of the graph.
Next, we graph g ๏จ x ๏ฉ ๏ฝ x ๏ญ 3 for inputs x ๏พ 2 .
Copyright ยฉ 2020 Pearson Education, Inc.
20
Chapter R Functions, Graphs, and Models
Choosing some values for x and evaluating the
function, we have:
f ๏จ x๏ฉ
x
๏จ x, f ๏จ x ๏ฉ๏ฉ
Creating the input โ output table, we have:
g ๏จ x๏ฉ
x
๏จ x, g ๏จ x ๏ฉ๏ฉ
๏จ3, 0๏ฉ
๏จ4,1๏ฉ
4
1
5
๏จ5, 2๏ฉ
2
The open circle indicates that ๏จ2, ๏ญ1๏ฉ is not part
3
0
of the graph.
๏ญ3
6
๏ญ1
๏ญ2
0
๏ญ3
1
๏ญ2
3
6
๏จ๏ญ3, 6๏ฉ
๏จ๏ญ1, ๏ญ2๏ฉ
๏จ0, ๏ญ3๏ฉ
๏จ1, ๏ญ2๏ฉ
๏จ3, 6๏ฉ
Since the input x ๏ฝ 2 is not defined on this part
of the graph, the point ๏จ2,1๏ฉ is not part of the
graph. The open circle indicates that ๏จ2,1๏ฉ is not
part of the graph.
๏ฌ x2 ,
for x ๏ผ 0
๏ฏ
72. Graph g ๏จ x ๏ฉ ๏ฝ ๏ญ๏ญ3,
for x ๏ฝ 0 .
๏ฏ๏ญ2 x ๏ซ 3, for x ๏พ 0
๏ฎ
for x ๏ฝ ๏ญ3
๏ฌ ๏ญ6,
74. Graph f ๏จ x ๏ฉ ๏ฝ ๏ญ 2
.
๏ฎ ๏ญ x ๏ซ 5, for x ๏น ๏ญ3
for x ๏ฝ 2
๏ฌ๏ญ7,
.
73. Graph f ๏จ x ๏ฉ ๏ฝ ๏ญ 2
๏ฎ x ๏ญ 3, for x ๏น 2
First, we graph f ๏จ x ๏ฉ ๏ฝ ๏ญ7 for x ๏ฝ 2 .
This graph consists of only one point, ๏จ2, ๏ญ7๏ฉ .
The solid dot indicates that ๏จ2, ๏ญ7๏ฉ is part of the
graph.
Next, we graph f ๏จ x ๏ฉ ๏ฝ x 2 ๏ญ 3 for inputs x ๏น ๏ญ2 .
Note that for f ๏จ x ๏ฉ ๏ฝ x 2 ๏ญ 3
4t
75.
๏ฆ 0.03 ๏ถ
A ๏จt ๏ฉ ๏ฝ P ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
We substitute 500 in for P and 2 in for t :
๏ฆ 0.03 ๏ถ
A ๏จt ๏ฉ ๏ฝ 500 ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
4๏ 2
๏ฝ 500 ๏จ1.0075๏ฉ
8
f ๏จ๏ญ3๏ฉ ๏ฝ ๏จ๏ญ3๏ฉ ๏ญ 3 ๏ฝ 6
2
๏ฝ 500 ๏จ1.061598848๏ฉ
f ๏จ ๏ญ1๏ฉ ๏ฝ ๏จ ๏ญ1๏ฉ ๏ญ 3 ๏ฝ ๏ญ2
2
๏ฝ 530.7994239
๏ป 530.80
The investment will be worth approximately
$530.80 after 2 years.
f ๏จ0๏ฉ ๏ฝ ๏จ0๏ฉ ๏ญ 3 ๏ฝ ๏ญ3
2
f ๏จ1๏ฉ ๏ฝ ๏จ1๏ฉ ๏ญ 3 ๏ฝ ๏ญ2
2
f ๏จ3๏ฉ ๏ฝ ๏จ3๏ฉ ๏ญ 3 ๏ฝ 6
2
We create the input โ output table.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.2
76.
21
๏ฆ 0.03 ๏ถ
A ๏จt ๏ฉ ๏ฝ P ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
80. a) Yes, the table represents a function. Each
event is assigned exactly one scale of
impact number.
b) The inputs are the events; the outputs are the
scale of impact numbers.
4t
๏ฆ 0.03 ๏ถ
A ๏จt ๏ฉ ๏ฝ 800 ๏ง1 ๏ซ
๏ท
๏จ
4 ๏ธ
4๏3
๏ฝ 800 ๏จ1.0075๏ฉ
12
๏ป 875.05
The investment will be worth approximately
$875.05 after 3 years.
hw
77. s ๏ฝ
3600
a) We substitute 170 for h and 70 for w.
๏จ170๏ฉ๏จ70๏ฉ
s๏ฝ
๏ป 1.818
3600
The patientโs approximate surface area is
1.818m2
b) We substitute 170 for h and 100 for w.
๏จ170๏ฉ๏จ100๏ฉ
s๏ฝ
๏ป 2.173
3600
The patientโs approximate surface area is
2.173m2
c) We substitute 170 for h and 50 for w.
๏จ170๏ฉ๏จ50๏ฉ
s๏ฝ
๏ป 1.537
3600
The patientโs approximate surface area is
1.537 m2
78.
s๏ฝ
81. Solve the equation for y.
2 x ๏ซ y ๏ญ 16 ๏ฝ 4 ๏ญ 3 y ๏ซ 2 x
y ๏ญ 16 ๏ฝ 4 ๏ญ 3 y
4 y ๏ฝ 20
y๏ฝ5
We sketch a graph of the equation.
No vertical line meets the graph more than
once. Thus, the equation represents a function.
82. First we solve the equation for y.
2 y 2 ๏ซ 3x ๏ฝ 4 x ๏ซ 5
2 y2 ๏ฝ x ๏ซ 5
subtract 3 x from both sides
x๏ซ5
y2 ๏ฝ
divide both sides by 2
2
x๏ซ5
take the square root of both sides
y๏ฝ๏ฑ
2
We sketch a graph of the equation.
hw
3600
a) s ๏ฝ
๏จ150๏ฉ๏จ70๏ฉ
๏ป 1.708
3600
The patientโs approximate surface area is
1.708m2 .
b) s ๏ฝ
๏จ180๏ฉ๏จ70๏ฉ
๏ป 1.871
3600
The patientโs approximate surface area is
1.871m2 .
79. a) Yes, each term has a specific monthly
payment.
b) Yes, each month payment has a specific
term.
c) No, the interest rate has multiple monthly
payments.
d) Yes, each monthly payment has a specific
interest rate.
We can see that a vertical line will intersect the
graph more than once; therefore, this is not a
function.
83. First we solve the equation for y.
๏จ4 y ๏ฉ ๏ฝ 64 x
4 ๏จy ๏ฉ ๏ฝ 4 x
2
3
2
3
3
3
3
3
y2 ๏ฝ x
y๏ฝ๏ฑ x
Copyright ยฉ 2020 Pearson Education, Inc.
22
Chapter R Functions, Graphs, and Models
Next, we sketch a graph of the equation.
Now, we are able to look at the table. We will
have to enter the appropriate values of x.
We can see that a vertical line will intersect the
graph more than once; therefore, this is not a
function.
84. First, we solve the equation for y.
๏จ3 y ๏ฉ ๏ฝ 72x
3
2
2
From the table we conclude:
f ๏จ๏ญ3๏ฉ ๏ฝ 2
9 y 3 ๏ฝ 72 x
y3 ๏ฝ 8x
f ๏จ ๏ญ2 ๏ฉ ๏ฝ 0
y ๏ฝ 3 8x
y ๏ฝ 23 x
๏ y ๏ณ 0๏
Note: since y must be non-negative to satisfy the
original equation, we only graph the points for
which y is non-negative.
Next, we sketch a graph of the equation:
No vertical line meets the graph more than
once. Thus, the equation represents a function.
85. Answers will vary. The vertical line test
works, because you are locating the values to
which each input corresponds. For a relation to
be a function, each input must correspond with
exactly one output. Therefore, if a vertical line
intersects the graph in more than one point, that
particular input corresponds to more than one
output and the graph is not a function.
86.
f ๏จ x๏ฉ ๏ฝ x ๏ญ 2 ๏ซ x ๏ซ 1 ๏ญ 5
f ๏จ 0 ๏ฉ ๏ฝ ๏ญ2
f ๏จ 4๏ฉ ๏ฝ 2
87.
f ๏จ x ๏ฉ ๏ฝ x3 ๏ซ 2 x 2 ๏ญ 4 x ๏ญ 13
88.
f ๏จ x๏ฉ ๏ฝ
3
2
x ๏ญ4
We begin by setting up the table:
Next, we will type in the equation into the
graphing editor.
First, we set up the table to allow us to ask the
calculator to compute specific values.
Next, we type the equation into the graphing
editor.
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.2
23
Now, we are able to look at the table:
89. Each graph is shown below.
In order to graph f ๏จ x ๏ฉ ๏ฝ x3 ๏ซ 2 x 2 ๏ญ 4 x ๏ญ 13 , we
use the window at the top of the next column.
After entering the function into the graphing
editor, we get:
90. Answers will vary. Some ordered pairs are
๏จ๏ญ2.978723,1.0617 ๏ฉ , ๏จ๏ญ1.382979, 2.8438301๏ฉ ,
๏จ0,3.1622777 ๏ฉ , ๏จ1.7021277, 2.6651006๏ฉ , and
๏จ2.6595745,1.7107494๏ฉ.
91. a) f ๏จ x ๏ฉ ๏ฝ 5( x ๏ซ 2)
After entering the function into the graphing
editor, we get:
In order to graph f ๏จ x ๏ฉ ๏ฝ
standard window:
3
2
x ๏ญ4
g ๏จ x๏ฉ ๏ฝ 5x ๏ซ 2
b)
, we use the
c) No, the graph is not the same function.
92. a) f ๏จ x ๏ฉ ๏ฝ ( x ๏ญ 4)2
g ๏จ x๏ฉ ๏ฝ x2 ๏ญ 4
After entering the function into the graphing
editor, we get:
In order to graph f ๏จ x ๏ฉ ๏ฝ x ๏ญ 2 ๏ซ x ๏ซ 1 ๏ญ 5 , we
use the standard window.
b)
c) No, the graph is not the same function.
Copyright ยฉ 2020 Pearson Education, Inc.
24
93.
Chapter R Functions, Graphs, and Models
f ๏จ x ๏ฉ ๏ฝ 3x ๏ซ 6
g ๏จ x ๏ฉ ๏ฝ 3( x ๏ซ h)
The functions represent the same function, so
set them equal to each other to find the value for
h.
3 x ๏ซ 6 ๏ฝ 3( x ๏ซ h)
3 x ๏ซ 6 ๏ฝ 3x ๏ซ 3h
6 ๏ฝ 3h
2๏ฝh
94.
f ๏จ x ๏ฉ ๏ฝ ( x ๏ซ 3) 2
g ๏จ x๏ฉ ๏ฝ x2 ๏ซ 6 x ๏ซ h
The functions represent the same function, so
set them equal to each other to find the value for
h.
( x ๏ซ 3) 2 ๏ฝ x 2 ๏ซ 6 x ๏ซ h
x2 ๏ซ 6 x ๏ซ 9 ๏ฝ x2 ๏ซ 6 x ๏ซ h
9๏ฝh
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.3
Exercise Set R.3
1.
๏ ๏ญ2, 4๏
3.
๏จ0,5๏ฉ
4.
๏ ๏ญ1, 2๏
5.
๏จ๏ญ9, ๏ญ5๏
6.
๏ ๏ญ9, ๏ญ4๏ฉ
7.
๏ x, x ๏ซ h ๏
8.
๏จ x, x ๏ซ h ๏
9.
๏ ๏ญ4, ๏ญ1๏ฉ ๏ ๏จ2,3๏
10.
๏จ๏ญ๏ฅ, 0๏ฉ ๏ ๏3, ๏ฅ๏ฉ
11.
๏ ๏ญ2, 2๏
13.
19.
๏ ๏ญ4, ๏ญ3๏ฉ ๏ ๏จ0,5๏
20.
๏จ๏ญ๏ฅ, ๏ญ2๏ฉ ๏ ๏1, 4๏ฉ
๏จ๏ญ1,3๏ฉ
2.
12.
25
๏จ๏ญ5,5๏ฉ
๏จ6, 20๏
14.
๏ ๏ญ4, ๏ญ1๏ฉ
15.
๏จ๏ญ3, ๏ฅ๏ฉ
16.
๏จ๏ญ๏ฅ, ๏ญ2๏
17.
๏จ๏ญ2,3๏
18.
๏ ๏ญ10, 4๏ฉ
21. a) First, we locate 1 on the horizontal axis and
then we look vertically to find the point on
the graph for which 1 is the first coordinate.
From that point, we look to the vertical axis
to find the corresponding y-coordinate, 3.
Thus, f ๏จ1๏ฉ ๏ฝ 3 .
b) The domain is the set of all x-values of the
points on the graph. The domain is
๏ป๏ญ3, ๏ญ1,1,3,5๏ฝ .
c) First, we locate 2 on the vertical axis and
then we look horizontally to find any points
on the graph for which 2 is the second
coordinate. One such point exists, ๏จ3, 2๏ฉ .
Thus the x-value for which f ๏จ x ๏ฉ ๏ฝ 2 is
x ๏ฝ 3.
d) The range is the set of all y-values of the
points on the graph. The range is
๏ป๏ญ2, 0, 2,3, 4๏ฝ .
22. a)
f ๏จ1๏ฉ ๏ฝ ๏ญ1 .
b) The domain is ๏ป๏ญ4, ๏ญ3, ๏ญ2, ๏ญ1, 0,1, 2๏ฝ .
c) The point on the graph with the second
coordinate 2 is ๏จ๏ญ2, 2๏ฉ . Thus the x-value for
which f ๏จ x ๏ฉ ๏ฝ 2 is x ๏ฝ ๏ญ2 .
d) The range is ๏ป๏ญ2, ๏ญ1, 0,1, 2,3, 4๏ฝ .
23. a) First, we locate 1 on the horizontal axis and
then we look vertically to find the point on
the graph for which 1 is the first coordinate.
From that point, we look to the vertical axis
to find the corresponding y-coordinate, 4.
Thus, f ๏จ1๏ฉ ๏ฝ 4 .
b) The domain is the set of all x-values of the
points on the graph. The domain is
๏ป๏ญ5, ๏ญ3,1, 2,3, 4,5๏ฝ .
Copyright ยฉ 2020 Pearson Education, Inc.
26
Chapter R Functions, Graphs, and Models
c) First, we locate 2 on the vertical axis and
then we look horizontally to find any points
on the graph for which 2 is the second
coordinate. Three such point exists,
๏จ๏ญ5, 2๏ฉ ; ๏จ๏ญ3, 2๏ฉ ; and ๏จ4, 2๏ฉ . Thus the xvalues for which f ๏จ x ๏ฉ ๏ฝ 2 are ๏ป๏ญ5, ๏ญ3, 4๏ฝ .
d) The range is the set of all y-values of the
points on the graph. The range is
๏ป๏ญ3, 2, 4,5๏ฝ
24. a)
f ๏จ1๏ฉ ๏ฝ 2 .
b) The domain is ๏ป๏ญ6, ๏ญ4, ๏ญ2, 0,1,3, 4๏ฝ .
c) The points on the graph with the second
coordinate 2 are ๏จ1, 2๏ฉ and ๏จ3, 2๏ฉ . Thus the
x-values for which f ๏จ x ๏ฉ ๏ฝ 2 are ๏ป1,3๏ฝ .
d) The range is ๏ป๏ญ5, ๏ญ2, 0, 2,5๏ฝ .
25. a) First, we locate 1 on the horizontal axis and
then we look vertically to find the point on
the graph for which 1 is the first coordinate.
From that point, we look to the vertical axis
to find the corresponding y-coordinate, -1.
Thus, f ๏จ1๏ฉ ๏ฝ ๏ญ1 .
b) The domain is the set of all x-values of the
points on the graph. These extend from -2 to
4. Thus, the domain is ๏ป x | ๏ญ2 ๏ฃ x ๏ฃ 4๏ฝ , or
in interval notation ๏ ๏ญ2, 4๏ .
c) First, we locate 2 on the vertical axis and
then we look horizontally to find any points
on the graph for which 2 is the second
coordinate. One such point exists, ๏จ3, 2๏ฉ .
Thus the x-value for which f ๏จ x ๏ฉ ๏ฝ 2 is
x ๏ฝ 3.
d) The range is the set of all y-values of the
points on the graph. These extend from -3 to
3. Thus, the range is ๏ป y | ๏ญ3 ๏ฃ y ๏ฃ 3๏ฝ , or, in
interval notation ๏ ๏ญ3,3๏ .
26. a)
27. a) First, we locate 1 on the horizontal axis and
then we look vertically to find the point on
the graph for which 1 is the first coordinate.
From that point, we look to the vertical axis
to find the corresponding y-coordinate, -2.
Thus, f ๏จ1๏ฉ ๏ฝ ๏ญ2 .
b) The domain is the set of all x-values of the
points on the graph. These extend from -4 to
2. Thus, the domain is ๏ป x | ๏ญ4 ๏ฃ x ๏ฃ 2๏ฝ , or,
in interval notation ๏ ๏ญ4, 2๏ .
c) First, we locate 2 on the vertical axis and
then we look horizontally to find any points
on the graph for which 2 is the second
coordinate. One such point exists, ๏จ๏ญ2, 2๏ฉ .
Thus the x-value for which f ๏จ x ๏ฉ ๏ฝ 2 are
x ๏ฝ ๏ญ2.
d) The range is the set of all y-values of the
points on the graph. These extend from -3 to
3. Thus, the range is ๏ป y | ๏ญ3 ๏ฃ y ๏ฃ 3๏ฝ , or in
interval notation ๏ ๏ญ3,3๏ .
28. a)
f ๏จ1๏ฉ ๏ป 2.25 .
b) The domain is ๏ป x | ๏ญ4 ๏ฃ x ๏ฃ 3๏ฝ or ๏ ๏ญ4,3๏ .
c) The point on the graph with the second
coordinate 2 appears to be ๏จ0, 2๏ฉ . Thus the
x-value for which f ๏จ x ๏ฉ ๏ฝ 2 is x ๏ฝ 0. .
d) The range is ๏ป y | ๏ญ5 ๏ฃ y ๏ฃ 4๏ฝ , or, ๏ ๏ญ5, 4๏ .
29. a) First, we locate 1 on the horizontal axis and
then we look vertically to find the point on
the graph for which 1 is the first coordinate.
From that point, we look to the vertical axis
to find the corresponding y-coordinate, 3.
Thus, f ๏จ1๏ฉ ๏ฝ 3 .
b) The domain is the set of all x-values of the
points on the graph. These extend from -3 to
3. Thus, the domain is ๏ป x | ๏ญ3 ๏ฃ x ๏ฃ 3๏ฝ , or,
in interval notation ๏ ๏ญ3,3๏ .
f ๏จ1๏ฉ ๏ป 2.5 .
b) The domain is ๏ป x | ๏ญ3 ๏ฃ x ๏ฃ 5๏ฝ or ๏ ๏ญ3,5๏ .
c) The point on the graph with the second
coordinate 2 appears to be ๏จ2.25, 2๏ฉ . Thus
the x-value for which f ๏จ x ๏ฉ ๏ฝ 2 is x ๏ฝ 2.25.
d) The range is ๏ป y |1 ๏ฃ y ๏ฃ 4๏ฝ , or, ๏1, 4๏ .
c) First, we locate 2 on the vertical axis and
then we look horizontally to find any points
on the graph for which 2 is the second
coordinate. Two such point exists,
๏จ๏ญ1.4, 2๏ฉ and ๏จ1.4, 2๏ฉ . Thus the x-values for
which f ๏จ x ๏ฉ ๏ฝ 2 are ๏ป๏ญ1.4,1.4๏ฝ .
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.3
27
d) The range is the set of all y-values of the
points on the graph. These extend from -5 to
4. Thus, the range is ๏ป y | ๏ญ5 ๏ฃ y ๏ฃ 4๏ฝ , or in
33.
interval notation ๏ ๏ญ5, 4๏ .
30. a)
f ๏จ1๏ฉ ๏ป 2 .
b) The domain is ๏ป x | ๏ญ5 ๏ฃ x ๏ฃ 4๏ฝ or ๏ ๏ญ5, 4๏ .
adding x to both sides
2๏ฝ x
Thus, 2 is not in the domain of f, while all other
real numbers are. The domain of f is
๏ปx | x is a real number and x ๏น 2๏ฝ ; or, in
c) The points on the graph with the second
coordinate 2 are all the points with the xvalue in the set ๏ป x |1 ๏ฃ x ๏ฃ 4๏ฝ . Thus the xvalues for which f ๏จ x ๏ฉ ๏ฝ 2 are
interval notation, ๏จ ๏ญ ๏ฅ, 2๏ฉ ๏ ๏จ 2, ๏ฅ ๏ฉ
๏ป x |1 ๏ฃ x ๏ฃ 4๏ฝ , or ๏1, 4๏ .
d) The range is ๏ป y | ๏ญ3 ๏ฃ y ๏ฃ 2๏ฝ , or ๏ ๏ญ3, 2๏ .
31. a) First, we locate 1 on the horizontal axis and
then we look vertically to find the point on
the graph for which 1 is the first coordinate.
From that point, we look to the vertical axis
to find the corresponding y-coordinate, 1.
Thus, f ๏จ1๏ฉ ๏ฝ 1 .
34.
b) The domain is the set of all x-values of the
points on the graph. These extend from -5 to
5. However, the open circle at the point
๏จ5, 2๏ฉ indicates that 5 is not in the domain.
35.
dividing both sides by 2
x๏ณ0
The domain of f is
๏ปx | x is a real number and x ๏ณ 0๏ฝ ; or, in interval
values for which f ๏จ x ๏ฉ ๏ฝ 2 are
๏ปx | 3 ๏ฃ x ๏ผ 5๏ฝ , or ๏3,5๏ฉ .
d) The range is the set of all y-values of the
points on the graph. The range is
๏ป๏ญ2, ๏ญ1, 0,1, 2๏ฝ
f ๏จ1๏ฉ ๏ฝ 2 .
b) The domain is ๏ป x | ๏ญ4 ๏ฃ x ๏ฃ 4๏ฝ or ๏ ๏ญ4, 4๏ .
c) The points on the graph with the second
coordinate 2 are all the points with the xvalue in the set ๏ป x | 0 ๏ผ x ๏ฃ 2๏ฝ . Thus the xvalues for which f ๏จ x ๏ฉ ๏ฝ 2 are
๏ปx | 0 ๏ผ x ๏ฃ 2๏ฝ , or ๏จ0, 2๏.
d) The range is ๏ป1, 2,3, 4๏ฝ .
f ๏จ x๏ฉ ๏ฝ 2 x
Since the function value cannot be calculated
when the radicand is negative, the domain is all
real numbers for which 2 x ๏ณ 0 . We find them
by solving the inequality.
2x ๏ณ 0
setting the radicand ๏ณ 0
c) First, we locate 2 on the vertical axis and
then we look horizontally to find any points
on the graph for which 2 is the second
coordinate. We notice all the points with xvalues in the set ๏ป x | 3 ๏ฃ x ๏ผ 5๏ฝ Thus the x-
32. a)
2
x๏ซ3
Set the denominator equal to 0 and solve.
x๏ซ3 ๏ฝ 0
x ๏ฝ ๏ญ3
The domain is
๏ปx | x is a real number and x ๏น ๏ญ3๏ฝ , or, in
f ๏จ x๏ฉ ๏ฝ
interval notation, ๏จ๏ญ ๏ฅ, ๏ญ3๏ฉ ๏ ๏จ ๏ญ3, ๏ฅ ๏ฉ
Thus, the domain is ๏ป x | ๏ญ5 ๏ฃ x ๏ผ 5๏ฝ , or in
interval notation ๏ ๏ญ5,5๏ฉ .
6
2๏ญ x
Since the function value cannot be calculated
when the denominator is equal to 0, we solve
the following equation to find those real
numbers that must be excluded from the domain
of f.
2๏ญ x ๏ฝ 0
setting the denominator equal to 0
f ๏จ x๏ฉ ๏ฝ
notation, ๏0, ๏ฅ ๏ฉ
36.
f ๏จ x๏ฉ ๏ฝ x ๏ญ 2
Solve x ๏ญ 2 ๏ณ 0
x๏ณ2
The domain of f is
๏ปx | x is a real number and x ๏ณ 2๏ฝ ; or, in
interval notation, ๏ 2, ๏ฅ ๏ฉ
Copyright ยฉ 2020 Pearson Education, Inc.
28
37.
Chapter R Functions, Graphs, and Models
f ๏จ x๏ฉ ๏ฝ x2 ๏ญ 2 x ๏ซ 3
42.
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
38.
43.
f ๏จ x๏ฉ ๏ฝ x2 ๏ซ 3
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
39.
f ๏จ x ๏ฉ ๏ฝ 3x ๏ซ 7
x๏ญ2
6 x ๏ญ 12
Since the function value cannot be calculated
when the denominator is equal to 0, we solve
the following equation to find those real
numbers that must be excluded from the domain
of f.
6 x ๏ญ 12 ๏ฝ 0
setting the denominator equal to 0
adding 12 to both sides
6 x ๏ฝ 12
dividing both sides by 6
x๏ฝ2
Thus, 2 is not in the domain of f, while all other
real numbers are. The domain of f is
๏ปx | x is a real number and x ๏น 2๏ฝ ; or, in
f ๏จ x๏ฉ ๏ฝ
3x ๏ญ 1
7 ๏ญ 2x
Since the function value cannot be calculated
when the denominator is equal to 0, we solve
the following equation to find those real
numbers that must be excluded from the domain
of f.
7 ๏ญ 2x ๏ฝ 0
setting the denominator equal to 0
f ๏จ x๏ฉ ๏ฝ
adding 2 x to both sides
7 ๏ฝ 2x
7
dividing both sides by 2
๏ฝx
2
7
Thus, is not in the domain of f, while all
2
other real numbers are. The domain of f is
7๏ผ
๏ฌ
๏ญ x | x is a real number and x ๏น ๏ฝ ; or, in
2๏พ
๏ฎ
7๏ถ ๏ฆ7 ๏ถ
๏ฆ
interval notation, ๏ง ๏ญ ๏ฅ, ๏ท ๏ ๏ง , ๏ฅ ๏ท
๏จ
2๏ธ ๏จ2 ๏ธ
interval notation, ๏จ ๏ญ ๏ฅ, 2๏ฉ ๏ ๏จ 2, ๏ฅ ๏ฉ
44.
40.
8
3x ๏ญ 6
Solve 3x ๏ญ 6 ๏ฝ 0
x๏ฝ2.
The domain of f is
๏ปx | x is a real number and x ๏น 2๏ฝ ; or, in
f ๏จ x๏ฉ ๏ฝ
interval notation, ๏จ ๏ญ ๏ฅ, 2๏ฉ ๏ ๏จ 2, ๏ฅ ๏ฉ
41.
f ๏จ x๏ฉ ๏ฝ x ๏ญ 4
2x ๏ญ 1
9 ๏ญ 2x
Solve 9 ๏ญ 2 x ๏ฝ 0
9
x๏ฝ
2
9
Thus, is not in the domain of f, while all
2
other real numbers are. The domain of f is
9๏ผ
๏ฌ
๏ญ x | x is a real number and x ๏น ๏ฝ ; or, in
2๏พ
๏ฎ
9๏ถ ๏ฆ9 ๏ถ
๏ฆ
interval notation, ๏ง ๏ญ ๏ฅ, ๏ท ๏ ๏ง , ๏ฅ ๏ท
๏จ
2๏ธ ๏จ2 ๏ธ
f ๏จ x๏ฉ ๏ฝ
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.3
45.
29
g ๏จ x๏ฉ ๏ฝ 4 ๏ซ 5x
Since the function value cannot be calculated
when the radicand is negative, the domain is all
real numbers for which 4 ๏ซ 5 x ๏ณ 0 . We find
them by solving the inequality.
4 ๏ซ 5x ๏ณ 0
setting the radicand ๏ณ 0
subtracting 4 from both sides
5 x ๏ณ ๏ญ4
4
x๏ณ๏ญ
dividing both sides by 5
5
The domain of g is
4๏ผ
๏ฌ
๏ญ x | x is a real number and x ๏ณ ๏ญ ๏ฝ ; or, in
5๏พ
๏ฎ
๏ฉ 4 ๏ถ
interval notation, ๏ช ๏ญ , ๏ฅ ๏ท
๏ซ 5 ๏ธ
49.
g ๏จ x๏ฉ ๏ฝ
2x
2
x ๏ญ 25
Since the function value cannot be calculated
when the denominator is equal to 0, we solve
the following equation to find those real
numbers that must be excluded from the domain
of g.
setting the denominator equal to 0
x 2 ๏ญ 25 ๏ฝ 0
x 2 ๏ฝ 25
adding 25 to both sides
x ๏ฝ ๏ฑ 25
taking the square root or both sides
x ๏ฝ ๏ฑ5
Thus, ๏ญ5 and 5 are not in the domain of g,
while all other real numbers are. The domain of
g is ๏ป x | x is a real number and x ๏น ๏ญ5, x ๏น 5๏ฝ ;
or, in interval notation,
๏จ๏ญ ๏ฅ, ๏ญ5๏ฉ ๏ ๏จ๏ญ5,5๏ฉ ๏ ๏จ5, ๏ฅ๏ฉ
46.
g ๏จ x ๏ฉ ๏ฝ 2 ๏ญ 3x
Solve 2 ๏ญ 3x ๏ณ 0
2 ๏ณ 3x
2
๏ณx
3
The domain of g is
2๏ผ
๏ฌ
๏ญ x | x is a real number and x ๏ฃ ๏ฝ ; or, in
3๏พ
๏ฎ
2๏น
๏ฆ
interval notation, ๏ง ๏ญ ๏ฅ, ๏บ
๏จ
3๏ป
47.
g ๏จ x๏ฉ ๏ฝ x2 ๏ญ 2 x ๏ซ 1
50.
interval notation, ๏จ๏ญ ๏ฅ, ๏ญ6๏ฉ ๏ ๏จ๏ญ6, 6๏ฉ ๏ ๏จ6, ๏ฅ ๏ฉ
51.
g ๏จ x ๏ฉ ๏ฝ 4 x3 ๏ซ 5 x 2 ๏ญ 2 x
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
2
x 2 ๏ฝ 36
x ๏ฝ ๏ฑ6
The domain of g is
๏ปx | x is a real number and x ๏น ๏ญ6, x ๏น 6๏ฝ ; or, in
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
48.
x ๏ญ1
x ๏ญ 36
2
Solve x ๏ญ 36 ๏ฝ 0
g ๏จ x๏ฉ ๏ฝ
52.
1
x2 ๏ซ 9
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
g ๏จ x๏ฉ ๏ฝ
x
x ๏ซ1
We can calculate the function value for all
values of x, so the domain is the set of all real
numbers ๏ก
g ๏จ x๏ฉ ๏ฝ
2
Copyright ยฉ 2020 Pearson Education, Inc.
30
53.
Chapter R Functions, Graphs, and Models
g ๏จ x๏ฉ ๏ฝ x ๏ซ 1 ๏ซ 6 ๏ญ 2 x
Since the function value cannot be calculated
when the radicand is negative, the domain is all
real numbers for which x ๏ซ 1 ๏ณ 0 and
6 ๏ญ 2 x ๏ณ 0 . We find them by solving the
inequality.
x ๏ซ1 ๏ณ 0
setting the radicand ๏ณ 0
x ๏ณ ๏ญ1
subtracting 1 from both sides
6 ๏ญ 2x ๏ณ 0
setting the radicand ๏ณ 0
๏ญ2 x ๏ณ ๏ญ6
subtracting 6 from both sides
x ๏ฃ 3 dividing both sides by -2
Thus, the domain of g is the union of these two
inequalities ๏ป x | x ๏ณ ๏ญ1 and x ๏ฃ 3๏ฝ ; or, in
interval notation, ๏ ๏ญ1,3๏
56. The graph crosses the line y ๏ฝ 1 at every integer
value of x. Therefore, the set of x-values for
which g ๏จ x ๏ฉ ๏ฝ 1 is
๏ปx | ๏ญ5, ๏ญ4, ๏ญ3, ๏ญ2, ๏ญ1, 0,1, 2,3, 4,5๏ฝ .
57. First, we locate 2 on the vertical axis and then
we look horizontally to find any points on the
graph for which 2 is the second coordinate. We
notice the point with x-value is 1.
58. First, we locate โ4 on the vertical axis and then
we look horizontally to find any points on the
graph for which the second coordinate is greater
than โ4. We notice all the points with x-values
in the set ๏ป x | ๏ญ4 ๏ฃ x ๏ผ 2๏ฝ Thus the x-values for
which G ๏จ x ๏ฉ ๏พ ๏ญ4 are
๏ปx | ๏ญ4 ๏ฃ x ๏ผ 2๏ฝ , or ๏ ๏ญ4, 2๏ฉ .
54.
g ๏จ x ๏ฉ ๏ฝ 2 x ๏ซ 3 ๏ญ 12 ๏ญ 5 x
Since the function value cannot be calculated
when the radicand is negative, the domain is all
real numbers for which 2 x ๏ซ 3 ๏ณ 0 and
12 ๏ญ 5 x ๏ณ 0 . We find them by solving the
inequality.
2x ๏ซ 3 ๏ณ 0
setting the radicand ๏ณ 0
2 x ๏ณ ๏ญ3
subtracting 3 from both sides
3
dividing both sides by 2
x๏ณ๏ญ
2
12 ๏ญ 5 x ๏ณ 0
setting the radicand ๏ณ 0
subtracting 12 from both sides
๏ญ5 x ๏ณ ๏ญ12
12
x๏ฃ
dividing both sides by -5
5
Thus, the domain of g is the union of these two
3
12 ๏ผ
๏ฌ
inequalities ๏ญ x | x ๏ณ ๏ญ and x ๏ฃ ๏ฝ ; or, in
2
5๏พ
๏ฎ
๏ฉ 3 12 ๏น
interval notation, ๏ช ๏ญ , ๏บ
๏ซ 2 5๏ป
59. First, we locate 0 on the vertical axis and then
we look horizontally to find any points on the
graph for which the second coordinate is 0. We
notice all the points with x-values in the set
๏ปx | x ๏ฝ ๏ญ3 and 4๏ฝ Thus the x-values for which
W ๏จ x ๏ฉ ๏ฝ 0 are ๏ญ3 and 4 .
60. First, we locate 0 on the vertical axis and then
we look horizontally to find any points on the
graph for which the second coordinate is greater
than or equal to 0. We notice all the points with
x-values in the set ๏ป x | ๏ญ6 ๏ฃ x ๏ผ 4๏ฝ Thus the x-
values for which k ๏จ x ๏ฉ ๏ณ 0 are
๏ปx | ๏ญ6 ๏ฃ x ๏ผ 4๏ฝ , or ๏ ๏ญ6, 4๏ฉ .
61. The domain is the number of hours that Karen
will work, which is 0 to 80 or ๏0,80๏ . The range
of the function is how much Karen will earn,
which is a minimum of P(0) ๏ฝ 40(0) ๏ฝ 0 and a
maximum of P(80) ๏ฝ 40(80) ๏ฝ 3200 . Therefore
the range of the function will be ๏ 0,3200๏ .
55. The graph of f lies on or below the x-axis when
f ๏จ x ๏ฉ ๏ฃ 0 , so we scan the graph from left to
right looking for the values of x for which the
graph lies on or below the x axis. Those values
extend from -1 to 2. So the set of x-values for
which f ๏จ x ๏ฉ ๏ฃ 0 is ๏ป x | ๏ญ1 ๏ฃ x ๏ฃ 2๏ฝ , or, in
interval notation, ๏ ๏ญ1, 2๏ .
62. The domain is the amount of money Marcus
will spend, which is 0 to 200 or ๏ 0, 200๏ . The
range of the function is how much Marcus will
pay in taxes, which is a minimum of
T (0) ๏ฝ 0.05(0) ๏ฝ 0 and a maximum of
T (200) ๏ฝ 0.05(200) ๏ฝ 10 . Therefore the range
of the function will be ๏0,10๏ .
Copyright ยฉ 2020 Pearson Education, Inc.
Exercise Set R.3
31
63. a) The domain in interval notation is ๏ 25,102๏ .
b) The range in interval notation is ๏ 0, 450๏ .
c) Answers will vary. Since we are looking for
the greatest increase in the incidence of
breast cancer, we are looking for the steepest
portion of the graph. It appears that the
greatest increase occurs from age 50 to age
60.
64. a) The graph extends from x ๏ฝ 0 to x ๏ฝ 92.3 ,
so the domain, in interval notation, of the
function N is ๏ 0,92.3๏
b) The graph extends from N ๏จ x ๏ฉ ๏ฝ 0 to
N ๏จ x ๏ฉ ๏ฝ 6 million. Therefore, the range,
in interval notation, of the function N is
๏0, 6, 000, 000๏ .
c) Answers will vary. We would target
the 50 year old to 60 year old age group,
because that is the age group that has the
most number of hearing-impaired
Americans.
65. a) f (4) ๏ฝ 5 ๏ซ 3.50(3) ๏ฝ 15.50 . The charge for
the first mile is $5, and then the charge for the
next 3 miles is $3.50 each, giving a fare of
$15.50 for a 4-mile trip.
b) f (4.25) ๏ฝ 5 ๏ซ 3.50(4) ๏ฝ 19 . The 0.25 mile is
considered part of a fourth additional mile after
the first mile is charged at the $5 rate, so the
total fare is $19.00 for a trip of at least 4 miles
up to and including 5 miles.
c) The fares will be $5, $8.50, $12, and so on,
in increments of $3.50, up to $36.50. Thus the
range is {5, 8.5, 12, 15.5, 19, 22.5, 26, 29.5, 33,
36.5}.
66. a) S (95) ๏ฝ 20 . It cost $20 to ship an order
totaling $95.
b) S (102) ๏ฝ 20 ๏ญ 5 ๏ฝ 15 . It cost $15 to ship an
order totaling $102, since $2 is considered a
part of an extra $20.
c) Orders totaling up to and including $120
will be shipped for $15, those totaling up to and
including $140 will be shipped for $10, those
totaling up to and including $160 will be
shipped for $5. Thus, and order totaling $160.01
or more will be shipped for free.
d) The shipping charges are reduced in
increments of $5. Thus, the range of S is {0, 5,
10, 15, 20}.
67. The domain of the function is the union of
where the numerator and denominator are
defined. Thus it would be where x is greater
than equal to 0 and 5 ๏ญ x is greater than 0.
x๏ณ0
5๏ญ x ๏พ 0
5๏พ x
The domain is the union of these sets which is
๏0,5๏ฉ .
68. The domain of the function is the union of
where the numerator and denominator are
defined. Thus it would be where 3 ๏ญ x is
greater than equal to 0 and x does not equal 0.
x๏น0
3๏ญ x ๏ณ 0
3๏ณ x
The domain is the union of these sets which is
๏จ๏ญ๏ฅ, 0๏ฉ ๏ ๏จ0,3๏ฉ .
69. The domain of the function is where the
denominator is defined. Thus it would be where
๏จ
๏ฉ
x x 2 ๏ญ 9 does not equal 0.
๏จ
๏ฉ
x x2 ๏ญ 9 ๏น 0
x( x ๏ญ 3)( x ๏ซ 3) ๏น 0
0, ๏ญ3,3 ๏น x
The domain is the union of these sets which is
๏จ๏ญ๏ฅ, ๏ญ3๏ฉ ๏ ๏จ๏ญ3, 0๏ฉ ๏ ๏จ0,3๏ฉ ๏ ๏จ3, ๏ฅ๏ฉ .
70. The domain of the function is where the
denominator is defined. Thus it would be where
๏จ
๏ฉ
x ๏จ x ๏ญ x ๏ญ 12๏ฉ ๏น 0
x x 2 ๏ญ x ๏ญ 12 does not equal 0.
2
x( x ๏ญ 4)( x ๏ซ 3) ๏น 0
0, 4, ๏ญ3 ๏น x
The domain is the union of these sets which is
๏จ๏ญ๏ฅ, ๏ญ3๏ฉ ๏ ๏จ๏ญ3, 0๏ฉ ๏ ๏จ0, 4๏ฉ ๏ ๏จ4, ๏ฅ๏ฉ .
71. Answers may vary. One example is
1
.
f ๏จ x๏ฉ ๏ฝ
2
x x ๏ญ4
๏จ
๏ฉ
72. Answers may vary. One example is
1
f ๏จ x๏ฉ ๏ฝ
.
x( x ๏ซ 1)(๏ญ7)
Copyright ยฉ 2020 Pearson Education, Inc.

## Document Preview (35 of 1356 Pages)

You are viewing preview pages of the document. Purchase to get full access instantly.

-37%

### Solution Manual for Calculus and Its Applications, 2nd Edition

$18.99 ~~$29.99~~Save:$11.00(37%)

24/7 Live Chat

Instant Download

100% Confidential

Store

##### Alexander Robinson

0 (0 Reviews)

## Best Selling

The World Of Customer Service, 3rd Edition Test Bank

$18.99 ~~$29.99~~Save:$11.00(37%)

Chemistry: Principles And Reactions, 7th Edition Test Bank

$18.99 ~~$29.99~~Save:$11.00(37%)

Test Bank for Hospitality Facilities Management and Design, 4th Edition

$18.99 ~~$29.99~~Save:$11.00(37%)

Test Bank for Strategies For Reading Assessment And Instruction: Helping Every Child Succeed, 6th Edition

$18.99 ~~$29.99~~Save:$11.00(37%)

Data Structures and Other Objects Using C++ 4th Edition Solution Manual

$18.99 ~~$29.99~~Save:$11.00(37%)

Solution Manual for Designing the User Interface: Strategies for Effective Human-Computer Interaction, 6th Edition

$18.99 ~~$29.99~~Save:$11.00(37%)