Contents Chapter R
Review of Prerequisites
R.1
Sets and the Real Number Line
1
R.2
Models, Algebraic Expressions, and Properties of Real Numbers
8
R.3
Integer Exponents and Scientific Notation
13
R.4
Rational Exponents and Radicals
20
R.5
Polynomials and Multiplication of Radicals
28
Problem Recognition Exercises: Simplifying Algebraic Expressions
35
R.6
Factoring
36
R.7
Rational Expressions and More Operations on Radicals
45
Chapter R Review Exercises
59
Chapter R Test
67
Chapter 1
Equations and Inequalities
1.1
Linear Equations and Rational Equations
71
1.2
Applications and Modeling with Linear Equations
85
1.3
Complex Numbers
96
1.4
Quadratic Equations
102
Problem Recognition Exercises: Simplifying Expressions Versus Solving Equations
118
1.5
Applications of Quadratic Equations
120
1.6
More Equations and Applications
127
1.7
Linear Inequalities and Compound Inequalities
143
1.8
Absolute Value Equations and Inequalities
152
Problem Recognition Exercises: Recognizing and Solving Equations and Inequalities
160
Chapter 1 Review Exercises
164
Chapter 1 Test
180
Chapter 1 Cumulative Review Exercises
186
Chapter 2
Functions and Graphs
2.1
The Rectangular Coordinate System and Graphing Utilities
189
2.2
Circles
202
2.3
Functions and Relations
212
2.4
Linear Equations in Two Variables and Linear Functions
220
2.5
Applications of Linear Equations and Modeling
234
Problem Recognition Exercises: Comparing Graphs of Equations
244
2.6
Transformations of Graphs
246
2.7
Analyzing Graphs of Functions and Piecewise-Defined Functions
258
2.8
Algebra of Functions and Function Composition
271
Chapter 2 Review Exercises
283
Chapter 2 Test
296
Chapter 2 Cumulative Review Exercises
300
Chapter 3
Polynomial and Rational Functions
3.1
Quadratic Functions and Applications
303
3.2
Introduction to Polynomial Functions
318
3.3
Division of Polynomials and the Remainder and Factor Theorems
329
3.4
Zeros of Polynomials
342
3.5
Rational Functions
359
Problem Recognition Exercises: Polynomial and Rational Functions
385
Polynomial and Rational Inequalities
387
Problem Recognition Exercises: Solving Equations and Inequalities
412
Variation
417
Chapter 3 Review Exercises
424
Chapter 3 Test
439
Chapter 3 Cumulative Review Exercises
448
3.6
3.7
Chapter 4
Exponential and Logarithmic Functions
4.1
Inverse Functions
451
4.2
Exponential Functions
460
4.3
Logarithmic Functions
471
Problem Recognition Exercises: Analyzing Functions
482
4.4
Properties of Logarithms
486
4.5
Exponential and Logarithmic Equations
493
4.6
Modeling with Exponential and Logarithmic Functions
511
Chapter 4 Review Exercises
521
Chapter 4 Test
531
Chapter 4 Cumulative Review Exercises
537
Chapter 5
Systems of Equations and Inequalities
5.1
Systems of Linear Equations in Two Variables and Applications
541
5.2
Systems of Linear Equations in Three Variables and Applications
559
5.3
Partial Fraction Decomposition
587
5.4
Systems of Nonlinear Equations in Two Variables
605
5.5
Inequalities and Systems of Inequalities in Two Variables
628
Problem Recognition Exercises: Equations and Inequalities in Two Variables
652
Linear Programming
656
Chapter 5 Review Exercises
668
Chapter 5 Test
685
Chapter 5 Cumulative Review Exercises
697
5.6
Section R.1
Chapter R Review of Prerequisites Section R.1 Sets and the Real Number Line 1. set
2. whole
3. natural
4. integers
5. set-builder
6. roster
7. rational
8. irrational
9. left
10. absolute value
5 , which is the 1 ratio of 5 and 1, so it is an element of . True.
d. 5 can be written as
22. a. 1, 2,3, does not include
b. 0,1, 2,3, does not include
11. a b or b a
1 . 3
False. c. , 2, 1,0,1, 2, does not include
12. base; exponent or power
1 . False. 3
13. square 14. 0; undefined
d.
15. 3 is an element of the set of natural numbers. 16.
1 . False. 3
1 is the ratio of 1 and 3, so it is an 3 element of . True.
23. a. 1, 2,3, does not include 0.25 . False.
2 is an element of the set of rational 5 numbers.
b. 0,1, 2,3, does not include 0.25 . False.
17. 3.1 is not an element of the set of integers. 18. is not an element of the set of rational numbers.
c. , 2, 1,0,1, 2, does not include
19. The set of integers is a proper subset of the set of real numbers.
d. 0.25 is a repeating decimal, so it is an element of . True.
20. The set of rational numbers is a proper subset of the set of real numbers.
24. a. , 2, 1,0,1, 2, does not include . False.
0.25 . False.
21. a. 1, 2,3, does not include 5 . False.
b. cannot be written as the ratio of two integers, so it is not an element of . False.
b. 0,1, 2,3, does not include 5 . False.
c. cannot be represented by a terminating decimal or by a repeating decimal, so it is an element of . True.
c. , 2, 1,0,1, 2, .This includes 5 . True.
d. is an element of , so it is an element of . True.
1
Chapter R Review of Prerequisites 25. The set given can be written in roster form as , 10, 5,0,5,10, . The number 25 is an element of this set. True.
35. a. The first set is not a subset of the second, because the set of whole numbers does not include negative integers. False. b. The first set is a subset of the second, because all whole numbers are also integers. True.
26. The set given can be written in roster form as 1, 2,3,,9 . The number 7 is not an element of this set. False.
n , so all 1 integers are also rational numbers. True.
36. a. Any integer n can be written as
27. The number 0.8 is a repeating decimal, so it is rational. It can be written as 0.88 , and 0.88 0.8 . False.
b. The first set is not a subset of the second, because there all elements in the set of rational numbers that are not integers. False.
45 , so it 100 is rational. The number 0.45 can be written as 0.4545 , and 0.45 0.4545 . True.
28. The number 0.45 can be written as
37. a. The first set is not a subset of the second, because irrational numbers are real numbers that cannot be represented as a ratio of integers, so a number cannot be both rational and irrational. False.
29. The set given can be written in roster form as 0, 2, 4,6, . The number 22 is an element of this set. True.
b. The first set is not a subset of the second, because irrational numbers are real numbers that cannot be represented as a ratio of integers, so a number cannot be both rational and irrational. False.
30. The set given can be written in roster form as , 8, 4, 0, 4,8, . The number 24 is an element of this set. True.
38. a. The first set is a subset of the second, because the real numbers are made up of rational and irrational numbers. True.
31. Irrational numbers are real numbers that cannot be represented as a ratio of integers, so a number cannot be both rational and irrational. False.
b. The first set is a subset of the second, because the real numbers are made up of rational and irrational numbers. True.
n , so all 1 integers are also rational numbers. True.
32. Any integer n can be written as
39. a. 6
b. 6
c. 12 , 6
33. a. The first set is a subset of the second, because all of the elements in the first set are elements of the second set. True.
d. 0.3,0.33, 0.9, 12,
b. The first set is not a subset of the second, because 0 is in the first set but not in the second set. False. 34. a. The first set is a subset of the second, because all of the elements in the first set are elements of the second set. True.
e.
5,
f.
5, 0.3, 0.33, 0.9, 12,
6
40. a. 1
b. The first set is not a subset of the second, because NM and TX are in the first set but not in the second set. False.
c. 0, 4 , 1
2
11 ,6 4
11 , 6, 4 6
b. 0, 1 d. 0, 4, 0.48,1,9.4
Section R.1
41. a 5
42. b 6
5? 9 3 5 5 5? 9 3 3 5 5 3 25 ? 27 False 15 15
43. 3c 9
44. 8d 16
55. 7, ; x | x 7
45. m 4 70
46. n 7 4
56. 2, ; x | x 2
e. f.
2
2
, 13
54.
, 0, 4, 0.48,1, 13,9.4
47. 3.14 ?
57. , 4.1 ; x | x 4.1
?
3.14 3.141592654 True
58. , 2.93 ; x | x 2.93
?
48. 7 7 ?
59. 6, 0 ; x | 6 x 0
?
60. 2, 8 ; x | 2 x 8
7 2.645751311 True
49. 6.7 6.7
6.7 ⱨ 6.7 True
61.
, 6
62.
, 4
63.
7 1 , 6 3
64.
4 7 , 3 4
65.
4,
66.
3,
?
50. 2.1 2.1
2.1ⱨ 2.1 True ?
6.15 6.15
51.
?
6.151515 6.155555 False ?
2.93 2.93
52.
?
2.933333 2.939393 False
53.
9 ? 11 7 8 9 8 ? 11 7 7 8 8 7 72 ? 77 False 56 56
3
67. x | 3 x 7
68. x | 4 x 1
69. x | x 6.7
70. x | x 3.2
3 71. x x 5
7 72. x x 8
Chapter R Review of Prerequisites 73. 6 6 6
74. 4 4 4
88. 8 2 10 10 or 2 8 10 10
75. 0 0
76. 1 1
89. 8 3 8 3 or
3 8 8 3
77.
2 2
2 2 2 2
90. 11 5 11 5 or
78.
6 6
6 6 6 6
91. 6 2 2 6 or 2 6 2 6
79. a. 3 3
92. 3 3 or 3 3
b. 3 3 3
93. a. 42 4 4 16 b. 4 4 4 16
80. a. for m 11, m 11 m 11
2
b. for m 11, m 11 m 11 11 m
c. 42 4 4 16 d.
81. a. for x 2, x 2 x 2
f.
82. a. for t 6, t 6 t 6 t 6
4 is not a real number.
94. a. 92 9 9 81
b. for t 6, t 6 t 6
b. 9 9 9 81 2
z 5 z 5 1 z 5 z 5
c. 92 9 9 81 d.
b. for z 5, z 5 z 5 z 5 1 1 z 5 z 5 z 5 84. a. for x 7,
42
e. 4 2
b. for x 2, x 2 x 2 x 2
83. a. for z 5,
5 11 11 5
93
e. 9 3 f.
7x 7x 1 7x 7x
9 is not a real number.
95. a. 3 8 2 b. 3 8 2
b. for x 7, 7x 7x 7x 1 1 7 x 7 x 7x
c. 3 8 3 8 2
85. 1 6 5 5 or 6 1 5 5 86. 2 9 7 7 or 9 2 5 7
d.
100 10
e.
100 is not a real number.
f. 100 10
87. 3 4 7 7 or 4 3 7 7
96. a. 3 27 3 b. 3 27 3
4
Section R.1
2 106. 5 4 6 2 8 31
c. 3 27 3 27 3
d.
49 7
5 4 6 6 31
e.
49 is not a real number.
5 4 6 36 31
2
5 4 6 5
f. 49 7
5 4 30 5 26 21
3
2 2 2 8 2 97. 3 3 3 3 27
107.
52 32 25 9 16 4
4 4 4 64 4 98. 5 5 5 5 125
108.
62 82 36 64 100 10
99. 0.2 0.2 0.2 0.2 0.2
109.
9 16 3 4 7 49
110.
8 125 2 5 7 343
3
2
4
0.0016 100. 0.1 0.1 0.1 0.1 0.1
3
3
2
3
2
3
3
4
2
2 7 2 2 7 111. 4 4 5 10 2 5 10
0.0001 101.
169 13 25 5
121 11 36 6
102.
4 7 4 10 10
3 4 10 9 4 100
103. 20 12 36 32 2 20 12 36 9 2
20 12 4 2 20 12 2 20 24 4
25
1 2 1 2 1 1 112. 6 6 9 4 3 2
200 4 48 200 192 8
6 12 3 5 18 2
2
1
200 22 48
2
4 9 9 1 100 25
1 104. 200 22 6 4 200 22 12 4 2
2 105. 6 12 3 1 6 18
2
4 1 9 1 6 4 9 9 4 9 4 6 36 36
5 6 36
6 12 3 25 18 6 12 3 7 6 12 21 6 9 3
1 6 5 5 1 36 6 6
5
Chapter R Review of Prerequisites
113. 9 6 3 7 8 25
4000 118. Wh 175 4000 16.1
9 6 4 8 25
4000 175 4016.1
9 6 4 8 25 9 6 4 25
119. V
9 10 5 927
8 2 4 3 5 9 8 2 4 2 9 8 2 4 2 9 8 12 3
124. A bracket is used if an endpoint to an interval is included in the set.
2 8
144 25 3 10 28
125. represents the set of integers. All integers belong to (the set of rational numbers). Therefore, is a subset of . However, 1 contains fractions and decimals, such as 2 and 0.2, that are not integers. Therefore, .
169 13 6 13 13 6 Undefined 0 116.
121 cc
123. A parenthesis is used if an endpoint to an interval is not included in the set.
84 4
122 52 3 10
3
122. All integers can be written as the integer itself over the integer 1.
8 2 6 3
115.
14.44 8
121. All terminating decimals can be written as a decimal fraction—that is, a fraction with a denominator that is a power of 10.
8 24 3 5 9
11 13 4 2
3
1.0 250 120. V 6.0 10.2 L 0.5 293
114. 8 2 4 2 5 5 9
3.8 8 2
2
173.6 lb
9 6 4 25
2
4 92 22 32 5 2 4 9 7 4 12 4 3 12 4
25 4 9 3 3
20 Undefined 0
4000 117. Wh 200 4000 5.5 4000 200 4005.5
126. The statement 1 indicates that the set containing the element 1 is a subset of . However, the statement 1 does not make sense, because the number 1 (not written in set brackets) does not represent a set. A quantity that is not a set cannot be a subset of another set. 127. If n 0 , then n n n n 0
2
128. If n 0 , then n n n n n n 2n
2
129. If n 0 , then n n n n 2n
119.5 lb
6
Section R.1 130. If n 0 , then n n n n n n 0
141.
131. If n 0 , then n n 132. If n 0 , then n n n 133. b 0
134. a 0
135. b 0
136. a 0
6744.25 142.
137. a is negative, b is positive, and c is negative. ab 2 So the sign of 3 can be represented by c
2 , which is positive. 3
37,741.81 143.
138. a is negative, b is positive, and c is negative. a 2c So the sign of 4 can be represented by b
2 , which is negative. 4 0.58
139. a is negative, b is positive, and c is negative. The sum of a and c must be negative. So the
144.
b a c can be represented by a2 3 , which is negative. 2 3
sign of
0.18 140. a is negative, b is positive, and c is negative. The even power of any nonzero number is
145. There are missing parentheses around the denominator.
a b b c can positive. So the sign of b be represented by . Also, 2
4
either a b or b c can equal zero. So the sign of the expression can also be 0 0 or 0 0 . represented by The sign is positive or zero.
7
Chapter R Review of Prerequisites Section R.2 Models, Algebraic Expressions, and Properties of Real Numbers 1. 5, 2 ; x | 5 x 2
b. m 0.04 x 2 3.6 x 49
m 0.04 50 3.6 50 49 2
2. 1, ; x | x 1
m 31 mpg c. m 0.04 x 2 3.6 x 49
2 9 9 2 or 9 2 9 2
3.
m 0.04 75 3.6 75 49 2
4. 6 6 or 6 6 5. True
6. False
7. False
8. True
12 9 9 9.
2
m 4 mpg The model breaks down. The value of 4 mpg is not possible. 26. a. P 1718t 2 82,000t 10,000
P 1718 22 82,000 22 10,000 2
2 3
P 982,000 bacteria
7 1
b. P 1718t 2 82,000t 10,000
12 9 3 2 3 2
12 6 2 3 2
10.
P 1718 36 82,000 36 10,000 2
6 6 12 18 3 6
102 82
16 4
2
P 735,000 bacteria
12 36 2 3
c. P 1718t 2 82,000t 10,000
6 12 54 66 11 6 6
100 64
4 2
2
36
2
2
P 1718 48 82,000 48 10,000 2
P 12,000 bacteria The model breaks down. It is impossible to have 12,000 bacteria.
6 3 4 2
27. a. E 27.45t 704.6
E 27.45 3 704.6
11. dependent
12. breakdown
13. term
14. coefficient
15. commutative
16. 0
17. 1
18. identity
19. x
20. distributive
21. reciprocal
22. associative
23. a b c
24. 0
28. a. W 150 1.2t
25. a. m 0.04 x 3.6 x 49
W 142.8 lb
E $622.25 b. E 27.45t 704.6
E 27.45 15 704.6 E $292.85 c. No. Eventually, the model would produce a negative expenditure. This is not possible.
W 150 1.2 6
2
m 0.04 35 3.6 35 49 2
m 28 mpg
8
Section R.2 43. a. C 159n 0.11159n
b. W 150 1.2t
W 150 1.2 12
C 159n 17.49n C 176.49n
W 135.6 lb c. No. Eventually the model would produce a negative weight or a weight too small to sustain life. This is not possible. 29. Tn Ts 16
30. K C 273.15
31. P W M
32. S P D
33. a. J C 1
b. C J 1
34. a. R F 9
b. F R 9
35. D 0.25P
36. t 0.20c
37. v 2 gh
38. h 8 t
b. C 176.49n
C 176.49 4 C $705.96 44. a. C 149n 0.16 149n 40
C 149n 23.84n 40 C 172.84n 40 b. C 172.84n 40
C 172.84 2 40 C $385.68 45. 12 x 2 y 5 xy 4 9.2 xy 3
12 x 2 y 5 xy 4 9.2 xy 3
39. a. C 0.12k 14.89
Terms: 12x y , xy 4 , 9.2xy 3 ; coefficients: 12, 1 , 9.2 2
b. C 0.12k 14.89
C 0.12 1200 14.89 C $158.89
5
46. 6c 2 d 6.4cd 5 d 6
6c 2 d 6.4cd 5 d 6
40. a. C 3.58n 13.50
Terms: 6c d , 6.4cd 5 , d 6 ; coefficients: 6, 6.4 , 1 2
b. 5000 2000 3000 The family used 3 thousand gal over the 2000-gal base. C 3.58n 13.50
47.
C 3.58 3 13.50 C $24.24 41. a. C 640m 200n 500
5 1 5 m m 5 Term: ; coefficient: 5 m
2 1 2 n n 2 Term: ; coefficient: 2 n
48.
b. C 640m 200n 500
C 640 12 200 3 500 C $8780
49. x y z 1x y z
42. a. C 105n 35 L 40
Term: x y z ; coefficient: 1
b. C 105n 35 L 40
50. m 3 n 1m 3 n
C 105 12 35 2 40
Term: m 3 n ; coefficient: 1
C $1370
51. 7 x x 7
9
Chapter R Review of Prerequisites 52. 9 z z 9
b.
53. 3 w w 3
5 4 1 4 5
The multiplicative inverse is 54. 11 p p 11
7 7 64. a. 0 9 9
1 1 55. y y 3 3 56. z
The additive inverse is
5 5 z 2 2
The multiplicative inverse is
58. c 4 5 c 4 5 c 9
b. 0 ? 1 The multiplicative inverse is undefined.
4 9 4 9 60. p p 1 p p 9 4 9 4
66. a. 1 1 0 The additive inverse is 1 .
61. a. 8 8 0 The additive inverse is 8.
b. 1 1 1 The multiplicative inverse is 1.
1 b. 8 1 8
67. a. 2.1 2.1 0 The additive inverse is 2.1 .
1 The multiplicative inverse is . 8
1 b. 2.1 1 2.1
62. a. 9 9 0 The additive inverse is 9 .
The multiplicative inverse is
1 b. 9 1 9
1 10 or . 2.1 21
68. a. 4.5 4.5 0 The additive inverse is 4.5 .
1 The multiplicative inverse is . 9
1 b. 4.5 1 4.5 The multiplicative inverse is 1 10 2 or . 4.5 45 9
5 5 0 4 4 The additive inverse is
9 . 7
65. a. 0 0 0 The additive inverse is 0.
1 1 5w 5 w 1w w 5 5
63. a.
7 . 9
7 9 b. 1 9 7
57. t 3 9 t 3 9 t 12
59.
4 . 5
5 . 4
69. 14 w3 3w3 w3 14 3 1 w3 16w3
10
Section R.2 70. 12t 5 t 5 6t 5 12 1 6 t 5 5t 5
77. 4 x 1 4 x 4 x
71. 3.9 x3 y 2.2 xy 3 5.1x 3 y 4.7 xy 3
3.9 x y 5.1x y 2.2 xy 4.7 xy 3
3
3
9 x 3 y 6.9 xy 3
3 2
4
3 2
24 x 2 16 x 8
0.004m n 0.01m n 0.005m n 4
4
3 2
80. 6 5 y 2 3 y 4
0.007 m3n 2
81.
82.
3 12 p 5 q 8 p 4 q 2 6 p 3q 4 3 3 3 12 p 5 q 8 p 4 q 2 6 p 3q 4 4 4 9 9 p 5 q 6 p 4 q 2 p 3q 2
8w 16 14w 28 12 22w 84. 3 2 z 4 8 z 9 84
6 z 12 8 z 72 84 14 z 85. 4u 8v 3 7u 2v 2v
4u 8v 21u 6v 2v 25u 16v
2
86. 10 x z 2 8 x 4 z 3 x
12 p 3 18.3 p 2 24 p 6.6 76. 2 5a 4 a 3 0.4a 0.8
10 x z 16 x 8 z 3x 29 x 9 z
87. 12 4 8 2v 5 3w 4v w
2 5a 2 a 2 0.4a 2 0.8 4
12 4 8 2v 15w 20v w
3
12 4 8 22v 15w w
10a 2a 0.8a 1.6 4
83. 2 4w 8 7 2w 4 12
3 4 p 3 6.1 p 3 8 p 3 2.2
75. 3 4 p 3 6.1 p 2 8 p 2.2
2 6 x 2 y 18 yz 2 2 z 3 3 2 2 2 6 x 2 y 18 yz 2 2 z 3 3 3 3 4 4 x 2 y 12 yz 2 z 3 3
1 4 3 4 3 yz y z yz 4 y 4 z 10 4 2 1 4 3 3 yz yz 4 y 4 z y 4 z 10 4 2 1 4 10 3 2 3 yz yz 4 y 4 z y 4 z 10 10 4 2 2 1 4 10 3 6 yz yz 4 y 4 z y 4 z 10 10 4 4 11 4 3 4 yz y z 10 4
3
30 y 2 18 y 24
1 1 2 73. c 7 d cd 7 c 7 d 2cd 7 3 2 5 1 7 2 7 1 c d c d cd 7 2cd 7 3 5 2 5 1 3 2 1 2 c 7 d c 7 d cd 7 2cd 7 5 3 3 5 2 2 5 7 6 7 1 7 4 7 c d c d cd cd 15 15 2 2 1 3 c 7 d cd 7 15 2
6 5 y 2 6 3 y 6 4
0.006m 4 n 0.002m3 n 2
74.
79. 8 3 x 2 2 x 1 8 3 x 2 8 2 x 8 1
72. 0.004m n 0.005m n 0.01m n 0.007 m n 4
1 1 1 78. k 7 1 k 7 k 7 2 2 2
3
3
12 32 88v 60 w w 88v 59w 20
11
Chapter R Review of Prerequisites 88. 6 2 9 z 6 y 8 y z 11
97. W 2 L 3
6 2 9 z 6 y 8 y 8 z 11
98. L 3S 5
6 2 2 y 17 z 11
99. E2009 E2008 489,000
6 4 y 34 z 11 4 y 34 z 5
100. C2011 C2010 0.25
2 89. 2 y 2 13 6 y 2 9 10 9 3
1 1 c h 15 25 1 1 G 240 500 15 25 G 16 20 36 gal
101. G
2 y 4 y 9 9
2 y 2 13 4 y 2 6 10 9 2
2
2 y2 4 y2 9 9 6 y2 3 90. 6 5t 2 12 8t 2 5 11t 2 4
1 1 c h 22 32 1 1 G 220 512 22 32 G 10 16 26 gal
102. G
6 11t 4 11t
6 5t 2 9 6t 2 5 11t 2 2
2
6 11t 2 4 11t 2 10
103. The commutative property of addition indicates that the order in which two quantities are added does not affect the sum. The associative property of addition indicates that the manner in which quantities are grouped under addition does not affect the sum.
b 6 6 3 91. 2a 2 1 2 92.
b 2 4ac
6 2 4 2 4
36 32 4 2 93.
104. For any nonzero number a, the multiplicative 1 inverse is defined as . If a 0 , this would a 1 be which is undefined. 0
x2 x1 2 y2 y1 2
1 2 2 1 4
32 52 9 25 34
2
105.
y y1 3.7 3.1 6.8 2 94. 2 x2 x1 2 1.4 3.4 95.
96.
1 2 1 22 2 r h 7 6 3 3 7 22 6468 49 6 308 21 21
106.
4 3 4 22 3 88 9 792 r 3 27 3 3 7 7 21 7
12
Section R.3 Section R.3 Integer Exponents and Scientific Notation 1.
2.
0
1 1 5 x 2 y 15 2 y 20 5 10 2 1 3 x y 3 y 2 x y 1 5 5 5
2 d. p 1 3 0
3 16. a. 1 7
1 5 3a 2b 6 12a 2b 3 6 2 5 7 a b 2 10a b 9a b 2 3 3 3
b.
30 1 1 1 30 1 7 7 7 7
3 3 3 3 c. w0 w0 1 7 7 7 7
3. a 6 a 6 6 a
0
3 d. w 1 7
4. 2 t 2 t t 2 5. t w 1.93
6. L D 3
7. 1
8. n
9. scientific
10. magnitude
11. m n
12. quotient
17. a. 82
1 1 2 8 64
b. 8 x 2 8
1 8 2 2 x x
c. 8 x
1
2
13. a. 8 1 0
1 64 x 2
1 1 1 d. 82 1 82 1 2 1 8 64 64
b. 80 1 80 11 1 c. 8 x 0 8 x 0 8 1 8
18. a. 7 2
d. 8 x 1 0
14. a. 70 1 b. 7 0 1 70 11 1
1 1 2 7 49
b. 7 y 2 7
1 7 2 2 y y
c. 7 y
1
2
c. 7 y 7 y 7 1 7 0
8x
2
0
d. 7 y 1
7 y
2
1 49 y 2
0
d. 7 2 1 7 2 1
1 1 1 1 2 7 49 49
0
2 15. a. 1 3 b.
19. a.
20 1 1 1 20 1 3 3 3 3
1 1 q2 q2 1 1 1 q 2 q2
b. q 2
2 2 2 2 c. p 0 p 0 1 3 3 3 3
13
1 q2
Chapter R Review of Prerequisites 1 5 p3 2 q2 q
30.
1 2 5q 2 q 3 p3 p
31.
c. 5 p 3 q 2 5 p 3 q 2 5 p 3 d. 5 p 3q 2 5 p 3 q 2 5
20. a.
1 t
4
1 t4 1 t 4 1 1 t4
1 b. t 4 4 t
32.
1 11u 2 c. 11t 4u 2 11 t 4 u 2 11 4 u 2 4 t t d. 11t 4u 2 11 t 4 u 2 11 t 4
1 11t 4 2 u2 u
21. 25 27 257 212 33. 22. 43 48 438 411 23. x 7 x 6 x 2 x 76 2 x11 24. y 3 y 7 y 4 y 37 4 y 8
25. 3c 2 d 7 4c 5 d 3 4 c 25 d 71
34.
12 c 3 d 8
72 1 1 7 24 7 2 2 4 7 7 49 18k 2 p 9 18 k 2 p 9 27 k 5 p 2 27 k 5 p 2 2 2 k 25 p 92 k 3 p 7 3 3 2 1 2 p7 3 p7 3 3 k 3k
10a 3b11 10 a 3 b11 25a 7b3 25 a 7 b3 2 2 a 37 b113 a 4 b8 5 5 8 2 1 2b 4 b8 4 5 a 5a 2m 6 n 4 2 m 6 n 4 6m 2 n 1 6 m 2 n 1 1 1 m 6 2 n 4 1 m 4 n5 3 3 5 1 1 n 4 n5 3 m 3m 4
4 p 7 q 5 4 p 7 q 5 2 2 2 p 7 2 q 5 2 2 2 2p q 2 p q 2 p 5 q 7 2
1 12d 8 12 3 d 8 3 c c
4 4
26. 7 m 3 n 8 3m 5 n 7 3 m 3 5 n 81
35. 42
21 m 8 n 7 1 1 21 21 8 7 8 7 m n mn
36. 23
27.
29.
23
6
2 2
37.
y 3 y 6 y 36 y 3 2 2 y 32 y1 y y2 y y z 8 z12 z 812 z 4 3 3 z 4 3 z1 z z3 z z
5
35
p p
15
2 7
27
p 14
q
42
q 8
38. q 4 28.
3
1 7 2q 7 q 5 p5 p
2
1 p14
1 q8
39. 2cd 2 c d 8c3 d 3 3
65 1 1 658 63 3 8 6 6 216
3
3
3
40. 8mn 8 m n 64m 2 n 2 2
14
2
2
2
Section R.3
7a 7 a 72 a 2 49a 2 7a 41. b b2 b2 b2 b2 2
2
2
36a 6b9 50. 9a 2b 4
2
2
4a 4b13
4 4 p4 q4 pq p4q4 pq 42. 4 256 256 4 4
4
43. 4 x 2 y 3
4 x y 16 x y 16yx 2
2 2
2
3 2
6
4
4
6
3 w z
3 5 2
44. 3w z
3 2
2
5 2
4 x 3 z 5 51. 12 y 2
9 z10 9w z 6 w 6 10
7k 45. 2 n
2
7 kn
5w 46. v 5
7 k n
2 2
2
2
3
4
3 z 2 w1 52. 15 p 4
5 3
3
1 3
3
2
1 4
3
4 a b 2
3
1 5
3
8 26
0
53. 2 y
64 64 1 1
3
1 16a8b 26
3
x y z 3 3
2 3
5 3
27 z15 x9 y 6 3
z w p 2 3
1 3
53 z 6 w3 p 12
1 8 2 43 1 2
13 2
a 8b 26
1 z 2 w1 p 4 5
1 1 v3 53 w15v3 3 15 v3 5 w 125w15 1 47. 8
4 2
33 x 9 y 6 z15
5 w v
5w v
2
1
1 3
1 1 4 n 2 n 49 k 49k 2 5 1 3
4 2
2
a b
2
1
1 x 3 y 2 z 5 3
4
3
4a 62b9 4
4 3
125w3 p12 z 6
6 y z 2 y 6 y z 3
2 8 2
3
2
2 2
8 2
2 62 y 3 y 4 z16 3
1 48. 9
2
1 3
4
0
1 92 34 1 27
2 62 y 7 z16 3
81 81 1 1 16m n 49. 5 2 8m n 2
7
2
2m 25 n7 2
2m 3 n9 2
2
1
2
2
54. 15 z
2
5z w 4
2 y
2
3
7
36 z16 9 z16 7 7 8 y 2y
6 3
w
15 z 2 53 z 4
m n 3 2
2
62 z16
3
6 3
15 53 w18 z 2 z12 2
9 2
15 53 w18 z10 2
m 6 n 18 2
m6
m6 22 n18 4n18
15
53 z10
125 z10 5 z10 152 w18 225w18 9w18
Chapter R Review of Prerequisites 1 1 1 55. 2 3 6
3
3 1 2 1 1 3 2 2 3 6
3
3
3
3 2 1 6 6 6 1 1 1 56. 3 4 2
2 6
3
x 3 4 x 2 61. x 6
1
x12 x 2 x 6
x16
1 3
2
4 3 6 12 12 12
2
1 12
2
y 12 y 3 y 4
y 11
2
1 3
3 2 2
1 6
3
1 58. 3
1 4
2
23 33 63
1 2
2
3 4 2 2
2
2
2
2
1
31 a 2b 4 6 12
3 a b 3
1
3 ab 2
5
v w x 15
10
20
x 32 16 x 32 v 21w28 16 1 v 21w28 2
3 2 3
5
3 2 1
3 6
9
6
1 5
3 2
4
1 4 4
4
16
4 16
143 v 6 w9 x 6 211 v 5 w3 x 2 7
4
v w x 16
4 16
143 21 7 v 15 w2 x8 4
1
3a b 3
1
14v w x 7v w x 64. 21v w x 14 v w x 7 v w x 21 v w x
16 y 4 2 y 4 7 8 x7 x 3
5
4281 x 7 y 4
1 3 60. 2 4 5 4 3a b a b
42 v 2 w6 x 4 5 1 v 15 w10 x 20 4 8 12 8 2 vw x
42 24 1 v 21w28 x 32
1
4 x 3 y 5 8 x13 y 14 2 6 10 1 13 14 4 x y 8 x y
4 5
2
42 v 2 w6 x 4 24 v 8 w12 x8 1
9 16 4 3 1 8 59. 3 5 13 14 4x y x y
1
3 2 4
2
8 27 216 197 2
1
4vw x v w x 63. 2v w x 3
1 57. 2
y11
2
3
1
1
y 2 6 y 3 62. y 4
12 144 3
1
x 16 1 16 x
33 27
4 1 3 1 6 1 4 3 3 4 6 2
5 4 1
143 21 x8
74 v15 w
65. 3 x 5
14
5 4
2
24 x8 v15 w2
3x 5 2 3x 514 2 3x 512
3 a b
66. 2 y 7 z
8
9a b8
16
4
2 y 7 z 13 2 y 7 z 413 9 2 y 7z
Section R.3 67. 6v 7 6v 7 10 9
109
68. 4 x 9 4 x 9 5 11
511
6v 7
c. Move the decimal point 0 places. The exponent on 10 is 0. 2.71 2.71 100
90
4 x 9
55
73. a. Move the decimal point 1 place. The number is between 0 and 1, so use a negative power of 10. 0.86 8.6 101
1 1 1 2 4 22 21 1 1 1 2 4 4 2 1 2 4 8 16 4 4 4 4 4 31 4
69. 22 21 20 21 22
b. Move the decimal point 0 places. The exponent on 10 is 0. 8.6 8.6 100 c. Move the decimal point 1 place. The number is greater than 10, so use a positive power of 10. 86 8.6 101
1 1 1 3 9 32 31 1 1 1 3 9 9 3 1 3 9 27 81 9 9 9 9 9 121 9
70. 32 31 30 31 32
74. a. Move the decimal point 1 place. The number is between 0 and 1, so use a negative power of 10. 0.792 7.92 101 b. Move the decimal point 0 places. The exponent on 10 is 0. 7.92 7.92 100 c. Move the decimal point 1 place. The number is greater than 10, so use a positive power of 10. 79.2 7.92 101
71. a. Move the decimal point 5 places. The number is greater than 10, so use a positive power of 10. 350, 000 3.5 105 b. Move the decimal point 5 places. The number is between 0 and 1, so use a negative power of 10. 0.000 035 3.5 10 5
75. Move the decimal point 10 places. The number is greater than 10, so use a positive power of 10. 29,980,000,000 cm/sec 2.998 1010 cm/sec
c. Move the decimal point 0 places. The exponent on 10 is 0. 3.5 3.5 100
76. Move the decimal point 10 places. The number is greater than 10, so use a positive power of 10. 149,000, 000 km 1.49 108 km
72. a. Move the decimal point 3 places. The number is greater than 10, so use a positive power of 10. 2710 2.71 103
77. Move the decimal point 5 places. The number is between 0 and 1, so use a negative power of 10. 0.000 01 cm 1.0 105 cm
b. Move the decimal point 3 places. The number is between 0 and 1, so use a negative power of 10. 0.002 71 2.71 10 3
78. Move the decimal point 12 places. The number is between 0 and 1, so use a negative power of 10. 0.000 000 000 001 sec 1.0 10 12 sec
17
Chapter R Review of Prerequisites 79. Move the decimal point 0 places. The exponent on 10 is 0. 4.2 L 4.2 100 L
c. 101 is greater than 10. Move the decimal point 1 place to the right. 1.87 101 18.7
80. Move the decimal point 0 places. The exponent on 10 is 0. 8.25 hr 8.25 100 hr
85. 1.67 1021 molecules
81. a. 10 6 is between 0 and 1. Move the decimal point 6 places to the left. 2.61 10 6 0.000 002 61
86. 3.0 1012 bytes 3, 000, 000, 000, 000 bytes
1,670,000,000,000,000,000,000 molecules
87. 7.0 106 m 0.000007 m
b. 106 is greater than 10. Move the decimal point 6 places to the right. 2.61 106 2, 610, 000
88. 4.7 10 7 0.000 000 47
89. 2 10 3 4 108 2 4 103 108
c. 10 1 The decimal point does not move. 2.61 100 2.61 0
8 10
5
90. 3 104 2 101 3 2 104 101
2
82. a. 10 is between 0 and 1. Move the decimal point 2 places to the left. 3.52 102 0.035 2
6 10
b. 102 is greater than 10. Move the decimal point 2 places to the right. 3.52 102 352
91.
8.4 106 8.4 106 2 4 104 2 2.1 10 2.1 10
c. 100 1 The decimal point does not move. 3.52 100 3.52
92.
6.8 1011 6.8 1011 2 108 3.4 103 3.4 103
3
93. 6.2 1011 3 104 6.2 3 1011 104
1
83. a. 10 is between 0 and 1. Move the decimal point 1 place to the left. 6.718 10 1 0.6718
18.6 10
15
1.86 101 1015 1.86 1016
b. 100 1 The decimal point does not move. 6.718 100 6.718
94. 8.1 106 2 105 8.1 2 106 105
c. 101 is greater than 10. Move the decimal point 1 place to the right. 6.718 101 67.18
16.2 1011
1.62 101 1011 1.62 10
12
84. a. 101 is between 0 and 1. Move the decimal point 1 place to the left. 1.87 101 0.187
95.
b. 100 1 The decimal point does not move. 1.87 100 1.87
3.6 1014 3.6 10 14 5 105 5 105 0.72 1019
7.2 101 10 19 7.2 10 20
18
Section 2.2 67. A circle is the set of all points in a plane that are equidistant from a fixed point called the center. 68. The center and radius can easily be identified from an equation of a circle written in standard form. 69. The calculator does not connect the pixels at the ends of the semicircles because of limited resolution.
73. 14, 39, 5 by 30, 5, 5
70. The screen is rectangular and is longer horizontally than vertically. If the x and y intervals are the same, then the graph will be elongated horizontally because there is greater distance horizontally over which to spread the graph. 71. 15.1,15.1,1 by 10,10, 1
74. 0.28, 0.18, 0.02 by 0.1, 0.2, 0.02
72. 15.1,15.1,1 by 10,10, 1
211
Chapter 2 Functions and Graphs
Section 2.3 Functions and Relations 1.
x2 x1 2 y2 y1 2
r
2
2 1 4 1 2 2
49 85 85 9 4 4 2
1 Center: , 2 ; Radius: 2
85 2
5. Substitute 0 for y: y x 3 5 x 2 4 x 20
2.
0 x 3 5 x 2 4 x 20 0 x 3 4 x 5 x 2 20
0 x x2 4 5 x2 4
0 x 4 x 5 0 x 2 x 2 x 5 x 2 or x 2 or x 5 Substitute 0 for x: y x 3 5 x 2 4 x 20
x x y y2 3. C 1 2 , 1 2 2
y 0 5 0 4 0 20 20
2 1 7 3 1 10 1 , , , 5 2 2 2 2 2 r
2
3
2
x-intercepts: 2, 0 , 2, 0 , 5, 0 ; y-intercept: 0, 20
x2 x1 2 y2 y1 2 2
1 2 2 7 5 2
6. Substitute 0 for y: y x3 3x 2 x 3 0 x3 3x 2 x 3
9 25 5 4 4 4 2 5 1 Center: , 5 ; Radius: 2 2
0 x3 3x 2 x 3 0 x3 x 3 x 2 3
0 x x2 1 3 x2 1
0 x 1 x 3
x x y y2 4. C 1 2 , 1 2 2
2
0 x 1 x 1 x 3 x 1 or x 1 or x 3 Substitute 0 for x: y x3 3x 2 x 3
3 4 5 1 1 4 1 , , 2 , 2 2 2 2 2
y 0 3 0 0 3 3 3
2
x-intercepts: 1, 0 , 1, 0 , 3, 0 ;
y-intercept: 0, 3
212
Section 2.3 7. relation; domain; y
25. No vertical line intersects the graph in more than one point. This relation is a function.
8. A relation defines y as a function of x if for each value of x in the domain, there is exactly one value of y in the range. 9. does not
10. 2, 4
11. y
12. f x
13. 5
26. There is at least one vertical line that intersects the graph in more than one point. This relation is not a function. 27. No vertical line intersects the graph in more than one point. This relation is a function. 28. There is at least one vertical line that intersects the graph in more than one point. This relation is not a function.
14. 5
15. a. {(Tom Hanks, 5), (Jack Nicholson, 12), (Sean Penn, 5), (Dustin Hoffman, 7)}
29. This mapping defines the set of ordered pairs: 8, 3 , 8, 4 , 8, 5 , 8, 6 . All
b. {Tom Hanks, Jack Nicholson, Sean Penn, Dustin Hoffman} c. {5, 12, 7}
four ordered pairs have the same x value but different y values. This relation is not a function.
d. Yes
16. a. {(Albany, 285), (Denver, 5883), (Miami, 11), (San Francisco, 11)}
30. This mapping defines the set of ordered pairs: 3,11 , 6,13 , 6, 9 . Two
b. {Albany, Denver, Miami, San Francisco} c. {285, 5883, 11}
ordered pairs have the same x value but different y values. This relation is not a function.
d. Yes
17. a. 4, 3 , 2, 3 , 1, 4 , 3, 2 , 3,1 b. 4, 2,1, 3
31. This mapping defines the set of ordered pairs: 1, 4 , 2, 5 , 3, 5 . No two ordered
c. 3, 3, 4, 2,1
pairs have the same x value but different y values. This relation is a function.
d. No 18. a. 2, 2 , 1, 0 , 0, 2 , 1, 0 , 1, 2 b. 2, 1, 0,1
32. This mapping defines the set of ordered pairs: 5, 4 , 6, 4 , 7, 4 , 8, 4 .
c. 2, 0, 2
No two ordered pairs have the same x value but different y values. This relation is a function.
d. No 19. False
20. True 33. x 1 y 5 25 This equation represents the graph of a circle with center 1, 5 and radius 5. This relation is not a function because it fails the vertical line test. 2
21. No vertical line intersects the graph in more than one point. This relation is a function. 22. No vertical line intersects the graph in more than one point. This relation is a function.
2
34. x 3 y 4 1 This equation represents the graph of a circle with center 3, 4 and radius 1. This relation is not a function because it fails the vertical line test.
23. There is at least one vertical line that intersects the graph in more than one point. This relation is not a function.
2
24. No vertical line intersects the graph in more than one point. This relation is a function.
213
2
Chapter 2 Functions and Graphs 35. y x 3
Two or more ordered pairs have the same x value but different y values. This relation is not a function.
38. a. y x y x
No vertical line intersects the graph in more than one point. This relation is a function.
36. y x 4 No vertical line intersects the graph in more than one point. This relation is a function.
b.
x y y x y x
No vertical line intersects the graph in more than one point. This relation is a function.
37. a. y x
b.
x
y
y x
Ordered pairs
0
0
y 0
1
1
y 1
2
2
y 2
3
3
y 3
0, 0 1, 1 2, 2 3, 3
2
Two or more ordered pairs have the same x value but different y values. This relation is not a function.
No vertical line intersects the graph in more than one point. This relation is a function.
39. 4,1
x y2
40. 7, 5
y2 x
41. f x x 2 3x
y x x
y
y x
0
0
y 00
1
1
y 1 1
4
2
y 4 2
9
3
y 9 3
a. f 2 2 3 2 4 6 2
Ordered pairs 0, 0
2
b. f 1 1 3 1 1 3 2 2
1, 1 4, 2 9, 3
c. f 0 0 3 0 0 0 0 2
d. f 1 1 3 1 1 3 4 2
e. f 2 2 3 2 4 6 10 2
214
Section 2.3 42. g x
49. k 5
1 x
5 1
4
Undefined
a. g 2
1 1 2 2
50. f 5 5 3 5 25 15 40
b. g 1
1 1 1
51. k 8
2
52. f 5 5 3 5 25 15 10 2
1 1 c. g 2 2 1 2
53. g t
1 1 d. g 2 2 1 2
e. g 2
1 1 t t
54. f a a 3 a a 2 3a 2
55. k a b
1 1 2 2
a b 1 a b 1
56. h a b 5
43. h x 5 a. h 2 5
57. f a 4 a 4 3 a 4 2
b. h 1 5
c. h 0 5
a 2 8a 16 3a 12
d. h 1 5
a 2 11a 28
e. h 2 5
58. f t 3 t 3 3 t 3 2
44. k x x 1 a. k 2
t 2 6t 9 3t 9 t 2 3t
2 1 1
1 0 Undefined
59. g 0
Undefined b. k 1
1 1 0 0
c. k 0
0 1 1 1
d. k 1
1 1 2
e. k 3
3 1 4 2
45. g 3
8 1 9 3
1 1 3 3
1 1 47. g 3 3 1 3
60. k 10
10 1
9
Undefined
46. h 7 5
61. f 9 7
62. f 1 6
63. f 3 4
64. f 2 3
65. f x 6 when x 1 66. f x 7 when x 9
48. h 7 5
67. f x 3 when x 2
215
Chapter 2 Functions and Graphs 68. f x 4 when x 3
d 2 18 2 36 Joe rides 36 mi in 2 hr.
12
75. Solve h x 0 :
Evaluate h 0 : h x x 8
2 2 d 18 12 3 3 12 mi
h x x 8
r x 250 x
x 8 x-intercepts: 8, 0 , 8, 0 ;
x 8
Evaluate k 0 :
0 x 2
k 0 0 2
k x x 2
x 2
2
x 2 x-intercepts: 2, 0 , 2, 0 ;
C 225 225 0.06 225 0.18 225
y-intercept: 0, 2
225 13.5 40.5 279 If the cost of the food is $225, then the total bill including tax and tip is $279.
77. Solve p x 0 :
Evaluate p 0 :
p x x 2 12
P x 1.075 x 0.40 x
p x x 2 12 p 0 0 12 12
0 x 2 12
P 60 1.075 60 0.40 60
2
x 2 12
1.075 60 24 1.075 84 90.30 If the cost of the book from the publisher is $60, the cost to the student after the bookstore markup and sales tax is $90.30.
0 2x 4
76. Solve k x 0 : k x x 2
C x x 0.06 x 0.18 x
f x 2 x 4
8
y-intercept: 0, 8
r x 250 x
73. Solve f x 0 :
h 0 0 8
0 x 8
r 122 250 122 128 128 mi
72.
g 0 3 0 12
x4 x-intercept: 4, 0 ; y-intercept: 0, 12
r 50 250 50 200 After having driven 50 mi, Frank still has 200 mi remaining.
71.
0 3x 12
g x 3x 12
3 x 12
40 2 b. 40 min is hr. 60 3 d t 18t
b.
Evaluate g 0 :
g x 3x 12
69. a. d t 18t
70. a.
74. Solve g x 0 :
x 12 2 3
x-intercepts: 2 3, 0 , 2 3, 0 ; y-intercept: 0,12
Evaluate f 0 :
78. Solve q x 0 :
f x 2 x 4
q x x 8 2
f 0 2 0 4
0 x2 8
2x 4
4 x2 x-intercept: 2, 0 ; y-intercept: 0, 4
x2 8 x 8 2 2
Evaluate q 0 : q x x2 8
q 0 0 8 8 2
x-intercepts: 2 2, 0 , 2 2, 0 ; y-intercept: 0, 8
216
Section 2.3 79. Solve r x 0 : r x x 8 0 x 8 x 8 0
84. Evaluate B 0 :
Evaluate r 0 : r x x 8
B x 0.12 x 2 0.86 x 1.6
r 0 0 8
B 0 0.12 0 0.86 0 1.6 1.6 2
0,1.6 ; The y-intercept means that for
8
x8 x-intercept: 8, 0 ; y-intercept: 0, 8
80. Solve s x 0 : s x x 3 0 x3 x3 0
x 0 (the year 2002), the number of Botox injection procedures in the United States was 1.6 million.
Evaluate s 0 :
85. Domain: 3, 2, 1, 2, 3 ;
s x x 3
Range: 4, 3, 3, 4, 5
s 0 0 3
86. Domain: 4, 2, 0, 2, 4 ;
3
x 3 x-intercept: 3, 0 ; y-intercept: 0, 3
81. Solve f x 0 :
Evaluate f 0 :
f x x 2
f x x 2
0 x 2 x2
f 0
Range: 2, 0, 2, 4, 6
87. Domain: 3, ; Range: 1, 88. Domain: , 4 ; Range: , 4
0 2
89. Domain: , ; Range: ,
2
x4 x-intercept: 4, 0 ; y-intercept: 0, 2
82. Solve g x 0 :
Evaluate g 0 :
g x x 3
g x x 3
0 x 3
g 0 0 3
x 3
90. Domain: , ; Range: , 91. Domain: , ; Range: 3, 92. Domain: , ; Range: , 2 93. Domain: 5,1 ; Range: 1,1, 3
3
x9 x-intercept: 9, 0 ; y-intercept: 0, 3
94. Domain: 4, 5 ; Range: 2,1, 4 95. x 4 0 x4 , 4 4,
83. Evaluate f 0 :
f x 9.4 x 35.7 f 0 9.4 0 35.7
96. x 2 0
35.7 0, 35.7 ; The y-intercept means that for x 0 (the year 2006), the average amount spent on video games per person in the United States was $35.70.
x2 , 2 2,
217
Chapter 2 Functions and Graphs 106. 6 x 0 x 6 x6 , 6
97. 2t 7 0 2t 7
t
7 2
7 7 , , 2 2
107. There is no restriction on x. ,
98. 3a 4 0 3a 4
a
108. There is no restriction on x. ,
4 3
109. a. f 2 4
4 4 , , 3 3
b. f 3 2 c. f x 1 for x 3 , x 1 , x 1
99. The denominator x 4 is always positive.
,
d. f x 4 for x 2 , x 2
10 e. x-intercepts: 0, 0 and , 0 3
100. The denominator x 2 3 is always positive. ,
f. y-intercept: 0, 0
101. a 15 0 a 15 15,
g. Domain: , h. Range: 4,
102. c 12 0 c 12 12,
110. a. f 2 1 b. f 3 3 c. There are no x-values where f x 1 .
103. 3 x 0 x 3
d. f x 4 for all x on the interval 0, 2
x3 , 3
e. x-intercept: 1, 0 f. y-intercept: 0, 4
104. 6 x 0 x 6 x6 , 6
g. Domain: , h. Range: 4 0, 111. a. f 2 0
105. 3 x 0 x 3 x3 , 3
b. f 3 5 c. f x 1 for x 3 , x 1 , x 1
218
Section 2.3 d. f x 4 for x 4 , x 0
119. f x 3x 2 2
4 e. x-intercepts: 2, 0 and , 0 3
120. f x x 3
f. y-intercept: 0, 4
121. If two points in a set of ordered pairs are aligned vertically in a graph, then they have the same x-coordinate but different ycoordinates. This contradicts the definition of a function. Therefore, the points do not define y as a function of x.
g. Domain: 4, h. Range: 4, 5 112. a. f 2 3
122. To find the x-intercept(s), find the real solutions to the equation f x 0 . To find
b. f 3 2
the y-intercept, evaluate f 0 .
c. f x 1 for x 0 , x 2
123. a. P s 4s
d. f x 4 for x 3 , x 5 e. x-intercept: 1, 0
b. A s s 2
f. y-intercept: 0, 1
P2 P c. A P or A P 4 16
2
g. Domain: , 5
d. P A 4 A
h. Range: 4, 0 3
e. d s s 2 s 2 2 s 2 2
113. r x x 400
d s 2s 2
r x 400 x
d s s 2
114. w x x 200
f. s d s d d 2 2
w x 200 x
2
2 s d d 2 2
115. P x x x x
2 d2 s d 2
P x 3x 116. 2 A x x 180
2 A x 180 x A x
180 x 2
sd
d2 2
sd
d
d 2 g. P d 4 2
117. C x x 90
C x 90 x
P d 2 2d
118. S x x 180
S x 180 x
219
2
or s d
d 2 2
Chapter 2 Functions and Graphs d 2 h. A d 2
2
d f. A d 2 A d
2d 2 A d 4 d2 A d 2
d 2 4
C g. A C 2
124. a. C r 2 r b. A r r 2 c. r d
2
d 2
AC
C 2 4 2
AC
C2 4
h. C A 2
d. d r 2r
2
A
2A C A 2 A C A 2
e. C d d
Section 2.4 Linear Equations in Two Variables and Linear Functions 1. a. No vertical line intersects the graph in more than one point. This relation is a function. b. ,
c. ,
d. 2, 0
e. 0, 3
c. ,
d. 3, 0
e. 0,1
c. 3
d. None
e. 0, 3
x x y y2 5. a. M 1 2 , 1 2 2 3 2 2 5 4 5 , 2 2
2. a. No vertical line intersects the graph in more than one point. This relation is a function. b. ,
b. ,
4 2 3 5 , 2 2 3 5 2 2, 3
3. a. No, a vertical line is not a function. b. 3
c. ,
d. 3, 0
e. None
b. d
x2 x1 2 y2 y1 2
2 3 2 4 5 5 2 2 5 5 2
2
4. a. No vertical line intersects the graph in more than one point. This relation is a function.
8 125 133
220
2
2
Section 2.4 20. 2 x y 4
x x y y2 6. a. M 1 2 , 1 2 2
y 2x 4
6 2 6 3 5 3 , 2 2
x 3 2 1 0
3 6 6 3 3 6 , , 3 3 2 2 2
b. d
x2 x1 2 y2 y1 2
2 6 6 5 3 3 6 4 3 2
2
y 2 0 2 4
2
x-intercept: 2, 0 ; y-intercept: 0, 4
2
21. 2 y 5 x 2
6 48 54 3 6
7. scatter
8. linear
5 y x 1 2
9. vertical
10. horizontal
x
11. True
12. m
y2 y1 x2 x1
1
13. zero
14. undefined
15. slope; intercept
16. mx b
17. m
f x2 f x1 x2 x1
0 2 5 2
18. horizontal
2
0 2
4
22. 3 y 4 x 6
4 y x2 3
4 y 3x 12
x 4
0
2 x-intercept: , 0 ; y-intercept: 0,1 5
19. 3 x 4 y 12 y
y 7 2 1
3 x3 4
x 1
y 0 3 2 3
0 3 2 3
9 2
y 10 3 2
0 2
3 x-intercept: , 0 ; y-intercept: 0, 2 2
x-intercept: 4, 0 ; y-intercept: 0, 3
221
Chapter 2 Functions and Graphs 23. x 6 x 6 6 6 6
26. 3x 2 4 3x 6 x2
y 2 0 2 4
x 2 2 2 2
y 2 0 2 4
x-intercept: 6, 0 ; y-intercept: None x-intercept: 2, 0 ; y-intercept: None
24. y 4 x 2 0 2 4
27. 0.02 x 0.05 y 0.1 2 x 5 y 10 5 y 2 x 10 2 y x2 5
y 4 4 4 4
x 5 0
x-intercept: None; y-intercept: 0, 4 25. 5 y 1 11
1
5 y 10 y2
5
x 2 0 2 4
y 4 2 8 5 0
x-intercept: 5, 0 ; y-intercept: 0, 2
y 2 2 2 2
28. 0.03x 0.07 y 0.21 3x 7 y 21 7 y 3x 21 3 y x3 7
x-intercept: None; y-intercept: 0, 2
x 1
0 1 7
y 24 7 3 18 7 0
x-intercept: 7, 0 ; y-intercept: 0, 3
222