-
ADDITIONAL ANSWERS 915
SECTION 1.7 Exploration 11.
2. 3. Linear: r2 � 0.9758; power: r2 � 0.9903; quadratic: R2 �
1; cubic: R2 � 1; quartic: R2 � 1
5. Since the quadratic curve fits the points perfectly, there is
nothing to be gained by adding a cubic term or a quartic term. The
coefficients of these terms in the regressions are zero.
Exercises 1.711. Let C be the total cost and n be the number of
items produced; C � 34,500 � 5.75n.12. Let C be the total cost and
n be the number of items produced; C � (1.09)28,000 � 19.85n.13.
Let R be the revenue and n be the number of items sold: R �
3.75n.14. Let P be the profit, and s be the amount of sales; then P
� 200,000 � 0.12s.21. x � 4x � 620; x � 124; 4x � 496 22. x � 2x �
3x � 714, so x � 119; the second and third numbers are 238 and
357.24. 179.9 25. 182 � 52t, so t � 3.5 hr 26. 560 � 45t � 55(t �
2), so t � 4.5 hours on local highways.27. 0.60(33) � 19.8,
0.75(27) � 20.25; The $33 shirt is a better bargain, because the
sale price is cheaper.31. (a) 0.10x � 0.45(100 � x) � 0.25(100) (b)
Use x � 57.14 gallons of the 10% solution and about 42.86 gal of
the 45% solution.32. 0.20x � 0.35(25 � x) � 0.26(25). Use x � 15
liters of the 20% solution and 10 liters of the 35% solution.34. 2x
� 2(x � 16) � 136. Two pieces that are x � 26 ft long are needed,
along with two 42 ft pieces.38. 900 � 0.07x � 0.085(12,000 � x);
$8000 was invested at 7%; the other $4000 was invested at 8.5%.41.
True; the correlation coefficient is close to 1 if there is a good
fit. 42. False; quadratic regression is useful for modeling
free-fall.47. (d) The point of intersection corresponds to the
break-even point, where C � R.49. (e) (f) You should recommend
stringing the rackets; fewer strung rackets need to
be sold to begin making a profit (since the intersection of y2
and y4 occurs for smaller x than the intersection of y1 and
y3).
[0, 10,000] by [0, 500,000]
[3, 11] by [0, 40]
n = 10; d = 35n = 9; d = 27n = 8; d = 20n = 7; d = 14
n = 6; d = 9n = 5; d = 5n = 4; d = 2
n = 3; d = 0
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 915
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50. (a) 51. (a)
CHAPTER 1 REVIEW EXERCISES11. (a) All reals (b) All reals 12.
(a) All reals (b) All reals 13. (a) All reals (b) [0, �) 14. (a)
All reals (b) [5, �)15. (a) All reals (b) [8, �) 16. (a) [�2, 2]
(b) [�2, 0] 17. (a) All reals except 0 and 2 (b) All reals except
0
18. (a) (�3, 3) (b) ��13�, �� 19. Continuous 20. Continuous 21.
(a) Vertical asymptotes at x � 0 and x � 5 (b) y � 023. (a) none
(b) y � 7 and y � �7 24. (a) x � �1 (b) y � 1 and y � �1 27. (��,
�1), (�1, 1), (1, �)33. (a) none (b) �7, at x � �134. (a) 2, at x �
�1 (b) �2, at x � 1 35. (a) �1, at x � 0 (b) none 36. (a) 1, at x �
2 (b) �1, at x � �245. 46. 47.
48. 49. 51.
52. f(x) � x � 3, x � �1; f(x) � x2 � 1, x � �1 53. ( f � g)(x)
� �x2� �� 4�; (��, �2] � [2, �)
54. (g � f )(x) � x � 4; [0, �) 55. ( f � g)(x) � �x�(x2 � 4);
[0, �) 56. ��gf��(x) � �g
f((xx))
� � ; [0, 2) � (2, �)
57. limx → �
�x� � � 58. limx → ��
�x2� �� 4� � �
65. (a) (b) The regression line is y � 61.133x � 725.333.
66. (a) (d)
[45, 110] by [–2, 18][45, 110] by [50, 70]
[4, 15] by [940, 1700][4, 15] by [940, 1700]
�x��x2 � 4
[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]
[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]
[0, 22] by [100, 200][–1, 15] by [9, 16]
916 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 916
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ADDITIONAL ANSWERS 917
67. (a) (b) 2�r2�3� �� r�2� (c) [0, �3�] 68. (c)(d) (e) 12.57
in.3
Chapter 1 Project1. 4. The actual growth in the number of
locations is
slowing while the model increases more rapidly.
5. y �
SECTION 2.1 Exploration 11. �$2000 per year 2. v(t) � �2000t �
50,000
Quick Review 2.11. y � 8x � 3.6 2. y � �1.8x � 2 3. y � �0.6x �
2.8 4. y � �
83
�x � �73
�
Exercises 2.11. Not a polynomial function because of the
exponent �5 2. Polynomial of degree 1 with leading coefficient 23.
Polynomial of degree 5 with leading coefficient 2 4. Polynomial of
degree 0 with leading coefficient 135. Not a polynomial function
because of the radical 6. Polynomial of degree 2 with leading
coefficient �5
y
6
x5
(1, 5)
(�2, �3)
y
7
x5
(3, 1)(�2, 4)
4914.198���1 � 269.459e�0.468x
[–1, 13] by [–100, 2600]
[0, 13] by [0, 20] [0, 6] by [0, 180]3
3 3h2r r
3 – r2h = 2
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 917
-
7. f(x) � �57
�x � �178� 8. f(x) � ��
79
�x � �83
� 9. f(x) � ��43
�x � �23
� 10. f(x) � �54
�x � �34
�
11. f(x) � �x � 3 12. f(x) � �12
�x � 2
19. Translate the graph of y � x2 3 units right and the 20.
Vertically shrink the graph of y � x2 by a factor of �14
�
result 2 units down. and translate the result down 1 unit.
21. Translate the graph of y � x2 2 units left, vertically 22.
Vertically stretch the graph of y � x2 by a factor of 3,
reflect
shrink the resulting graph by a factor of �12
�, and translate the result across the x-axis, and then
translate up 2 units.
that graph 3 units down.
23. Vertex: (1, 5); axis: x � 1 24. Vertex: (�2, �1); axis: x �
�2 25. Vertex: (1, �7); axis: x � 1
26. Vertex: (�3�, 4); axis: x � �3� 27. Vertex: ���56�,
��7132��; axis: x � ��56�; f(x) � 3�x � �
56
��2
� �7132�
28. Vertex: ��74�, �285��; axis: x � �74�; f(x) � �2�x � �
74
��2
� �285� 29. Vertex: (4, 19); axis: x � 4; f(x) � �(x � 4)2 �
19
30. Vertex: ��14�, �243��; axis: x � �14�; f(x) � 4�x � �
14
��2
� �243� 31. Vertex: ��35�, �
151��; axis: x � �35�; g(x) � 5�x � �
35
��2
� �151�
x
y
10
10
x
y
10
10
x
y
10
10
x
y
10
10
y
10
x10(–4, 0)
(0, 2)
y
5
x5
(0, 3)
(3, 0)
y
10
x10
(1, 2)
(5, 7)
y
10
x10
(–4, 6)
(–1, 2)
y
10
x10
(–3, 5)
(6, –2)
y
5
x3
(2, 4)
(–5, –1)
918 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 918
-
32. Vertex: ���74�, �187��; axis: x � ��74�; h(x) � �2�x � �
74
��2
� �187�
33. f(x) � (x � 2)2 � 2; Vertex: (2, 2); 34. g(x) � (x � 3)2 �
3; Vertex: (3, 3); 35. f(x) � �(x � 8)2 � 74; axis: x � 2; opens
upward; axis: x � 3; opens upward; Vertex: (�8, 74); axis: x � �8;
opens does not intersect x-axis does not intersect x-axis downward;
intersects x-axis at about
�16.602 and 0.602, or (�8 � �74� )
36. h(x) � �(x � 1)2 � 9; 37. f(x) � 2�x � �32��2
� �52
�; 38. g(x) � 5�x � �52��2
� �747�;
Vertex: (1, 9); axis: x � 1; Vertex: ���32�, �52
��; axis: x � ��32�; Vertex: ��52
�, ��747��; axis: x � �52�; opens
opens downward; intersects opens upward; does not intersect
upward; intersects x–axis at about
x-axis at �2 and 4 x-axis; vertically stretched by 2 0.538 and
4.462, or ��52� � �110��385��;
vertically stretched by 5
49. (a) (b) Strong positive 50. (a) (b) Strong negative
53. (a) y � 0.541x � 4.072. The slope tells us that hourly
compensation for production workers increases about 54¢/yr. (b)
About $25.7055. (a) [0, 100] by [0, 1000] is one possibility.58.
(b) 59. (b) (c) 90 cents per can; $16,200
61. (a) About 215 ft above the field (c) About 117 ft/sec
downward
[0, 15] by [10,000, 17,000][0, 25] by [200,000, 260,000]
[0, 90] by [0, 70][15, 45] by [20, 50]
[–5, 10] by [–20, 100]
[–3.7, 1] by [2, 5.1][–9, 11] by [–100, 10]
[–20, 5] by [–100, 100]
[–2, 8] by [0, 20] [–4, 6] by [0, 20]
919 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 919
-
63. (a) h � �16t2 � 80t � 10 64. 32�3� or about 55.426 ft/sec67.
(a) (c) On average, the children gained
0.68 pounds per month.(d)
68. (a) y � 548.30x � 21027.56 (b) About $59,400
69. The Identity Function f(x) � x
Domain: (��, �); Range: (��, �); Continuous; Increasing for all
x; Symmetric about the origin; Not bounded; No local extrema; No
horizontal or vertical asymptotes; End behavior: lim
x → ��f(x) � ��, lim
x → �f(x) � �
70. The Squaring Function f(x) � x2
Domain: (��, �); Range: [0, �); Continuous; Increasing on [0,
�), decreasing on (��, 0]; Symmetric about the y-axis; Bounded
below, but not above; Local minimum of 0 at x � 0; No horizontal or
vertical asymptotes; End behavior: lim
x → ��f(x) � lim
x → �f(x) � �
72. True. We can rewrite f in the form f(x) � �x � �14��2
� �34
�, so f � 0.
80. (a) (b) y � 0.115x � 8.245 (c) y � 0.556x � 6.093 (d) The
median–median line appears to be the better fit, becauseit
approximates more of the data val-ues more closely.
81. (a) The two solutions are��b � �
2ab2 � 4�ac�� and �
�b � �2a
b2 � 4�ac��; their sum is 2���2
ba�� � ��ba�.
(b) The product of the two solutions given above is � �ac
�.
82. f(x) � (x � a)(x � b) � x2 � (a � b)x � ab; the axis is
given by x � �[�(a � b)]/2, or x � (a � b)/2.
83. ��a �2b
�, ��(a �
4b)2
�� 84. x1 and x2 are given by the quadratic formula ��b � �2ab2
� 4�ac��; then x1 � x2 � ��
ba
�, and the line
of symmetry is x � ��2ba�, which is exactly equal to �
x1 �2
x2�.
b2 � (b2 � 4ac)��
4a2
[0, 15] by [0, 15][0, 17] by [2, 16]
[0, 17] by [2, 16]
[–4.7, 4.7] by [–1, 5]
[–4.7, 4.7] by [–3.1, 3.1]
[15, 45] by [20, 40]
[15, 45] by [20, 40][0, 5] by [–10, 100]
920 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 920
-
SECTION 2.2 Exploration 11. The pairs (0, 0), (1, 1) and (�1,
�1) are
common to all three graphs.
2. The pairs (0, 0), (1, 1), and (�1, 1) are common toall three
graphs.
Exercises 2.2
1. power � 5, constant � ��12
� 2. power � �53
�, constant � 9 3. not a power function 4. power � 0, constant �
13
5. power � 1, constant � c2 6. power � 5, constant � �2k
� 7. power � 2, constant � �g2
�
8. power � 3, constant � �43�� 9. power � �2, constant � k
10. power � 1, constant � m 11. degree � 0, coefficient � �4 12.
not a monomial function; negative exponent13. degree � 7,
coefficient � �6 14. not a monomial function: variable in exponent
15. degree � 2, coefficient � 4�16. degree � 1, Coefficient � l 23.
The weight w of an object varies directly with its mass m, with the
constant of variation g.24. The circumference C of a circle is
proportional to its diameter D, with the constant of variation
�.25. The refractive index n of a medium is inversely proportional
to v, the velocity of light in the medium, with constant of
variation c, the constant velocity of light in free space. 26. The
distance d traveled by a free-falling object dropped from rest
varies directly with the square
of its speed p, with the constant of variation �21g�.
27. power � 4, constant � 2; Domain: (��, �); 28. power � 3,
constant � �3; Domain: (��, �); Range: [0, �); Continuous;
Decreasing on (��, 0). Range: (��, �); Continuous; Decreasing for
all x; Odd. Increasing on (0, �); Even. Symmetric with respect
Symmetric with respect to origin; to y-axis; Bounded below, but not
above; Not bounded above or below; Local minimum at x � 0;
Asymptotes: none; No local extrema; Asymptotes: none; End Behavior:
lim
x → ��2x4 � �, lim
x → �2x4 � � End Behavior: lim
x → ���3x3 � �, lim
x → ��3x3 � ��
[–5, 5] by [–20, 20][–5, 5] by [–1, 49]
[–1.5, 1.5] by [–0.5, 1.5] [–5, 5] by [–5, 25] [–15, 15] by
[–50, 400]
[–5, 5] by [–15, 15] [–20, 20] by [–200, 200][–2.35, 2.35] by
[–1.5, 1.5]
ADDITIONAL ANSWERS 921
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 921
-
29. power � �14
�, constant � �12
�; Domain: [0, �); 30. power � �3, constant � �2; Domain: (��,
0) � (0, �);
Range: [0, �); Continuous; Increasing on [0, �); Range: (��, 0)
� (0, �); Discontinuous at x � 0; Bounded below; Neither even nor
odd; Increasing on (��, 0) and on (0, �).; Odd. Symmetric Local
minimum at (0, 0); Asymptotes: none; with respect to origin; Not
bounded above or below;
End Behavior: limx → �
�12
��4 x� � � No local extrema; Asymptotes at x � 0 and y � 0.; End
Behavior: lim
x → ���2x�3 � 0, lim
x → ��2x�3 � 0.
31. shrink vertically by �23
�; f is even. 32. stretch vertically by 5; f is odd. 33. stretch
vertically by 1.5 and reflect
over the x-axis; f is odd.
34. stretch vertically by 2 and reflect 35. shrink vertically by
�14
�; f is even. 36. shrink vertically by �18
�; f is odd.over the x-axis; f is even.
43. k � 3, a � �14
�. f is increasing in Quadrant I. f is undefined for x 0. 44. k
� �4, a � �23
�. f is decreasing in Quadrant IV.
f is even. 45. k � �2, a � �43
�. f is decreasing in Quadrant IV. f is even. 46. k � �25
�, a � �52
�. f is increasing in Quadrant I.
f is undefined for x 0. 47. k � �12
�, a � �3. f is decreasing in Quadrant I. f is odd.
48. k � �1, a � �4. f is increasing in Quadrant IV. f is
even.
49. y � �x82�, power � �2, constant � 8 50. y � �2�x�, power �
�
12
�, constant � �2 54.
[–5, 5] by [–19, 1][–5, 5] by [–50, 50][–5, 5] by [–1, 49]
[–5, 5] by [–20, 20]
[–5, 5] by [–20, 20][–5, 5] by [–1, 19]
[–5, 5] by [–5, 5][–1, 99] by [–1, 4]
922 ADDITIONAL ANSWERS
Wind Speed (mph) Power (W)
10 0015
0120
40 0960
80 7680
20
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 922
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55. (a) (c) (d) Approximately 37.67 beats/min, which is very
close to Clark’s observed value
57. (a) (c) (d) Approximately 2.76 �mW
2�
and 0.697 �mW
2�, respectively
59. False, because f(�x) � (�x)1/3 � �x1/3 � �f(x). The graph of
f is symmetric about the origin.65. (a) The graphs of f(x) � x�1
and h(x) � x�3
are similar and appear in the 1st and 3rdquadrants only. The
graphs of g(x) � x�2 andk(x) � x�4 are similar and appear in the
1stand 2nd quadrants only. The pair (1, 1) iscommon to all four
functions.
[–2, 2] by [–2, 2][0, 3] by [0, 3][0, 1] by [0, 5]
[0.8, 3.2] by [�0.3, 9.2][0.8, 3.2] by [–0.3, 9.2]
[–2, 71] by [50, 450][–2, 71] by [50, 450]
ADDITIONAL ANSWERS 923
f g h k
Domain x 0 x 0 x 0 x 0
Range y 0 y � 0 y 0 y � 0
Continuous yes yes yes yes
Increasing (��, 0) (��, 0)
Decreasing (��, 0), (0, �) (0, �) (��, 0), (0, �) (0, �)
Symmetry w.r.t. origin w.r.t. y-axis w.r.t. origin w.r.t.
y-axis
Bounded not below not below
Extrema none none none none
Asymptotes x-axis, y-axis x-axis, y-axis x-axis, y-axis x-axis,
y-axis
End Behavior limx → ��
f(x) � 0 limx → ��
g(x) � 0 limx → ��
h(x) � 0 limx → ��
k(x) � 0
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 923
-
(b) The graphs of f(x) � x1/2 and h(x) � x1/4
are similar and appear in the 1st quadrantonly. The graphs of
g(x) � x1/3 and k(x) � x1/5 are similar and appear in the 1stand
3rd quadrants only. The pairs (0, 0),(1, 1) are common to all four
functions.
SECTION 2.3Quick Review 2.33. (3x � 2)(x � 3) 5. x(3x � 2)(x �
1)8. x � 0, x � �2, x � 5
Exercises 2.31. Shift y � x3 to the right by 3 units, 2. Shift y
� x3 to the left by 5 units 3. Shift y � x3 to the left by 1
unit,
stretch vertically by 2. and then reflect over the x-axis.
vertically shrink by �12
�, reflect over the y-intercept: (0, �54) y-intercept: (0,
�125)
x-axis, and then vertically shift up
2 units. y-intercept: �0, �32��
x
y
5
5
x
y
15
200
x
y
10
10
[–3, 3] by [–2, 2][0, 3] by [0, 2][0, 1] by [0, 1]
924 ADDITIONAL ANSWERS
f g h k
Domain [0, �) (��, �) [0, �) (��, �)
Range y � 0 (��, �) y � 0 (��, �)
Continuous yes yes yes yes
Increasing [0, �) (��, �) [0, �) (��, �)
Decreasing
Symmetry none w.r.t. origin none w.r.t. origin
Bounded below not below not
Extrema min at (0, 0) none min at (0, 0) none
Asymptotes none none none none
End behavior
limx → �
f(x) � � limx → �
g(x) � �
limx → ��
g(x) � ��
limx → �
h(x) � � limx → �
k(x) � �
limx → ��
k(x) � ��
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 924
-
4. Shift y � x3 to the right by 3 units, 5. Shift y � x4 to the
left 2 units, 6. Shift y � x4 to the right 1 unit,
vertically shrink by �23
�, and vertically vertically stretch by 2, reflect over
vertically stretch by 3, and vertically
shift up 1 unit. y-intercept: (0, �17)the x-axis, and vertically
shift down shift down 2 units. y-intercept: (0, 1)3 units.
y-intercept: (0, �35)
7. local maximum: � (0.79, 1.19), zeros: x � 0 and x � 1.26.8.
local maximum: (0, 0), local minima: � (1.12, �3.13) and (�1.12,
�3.13), zeros: x � 0, x � 1.58, x � �1.58.13. One possibility: 14.
One possibility: 15. One possibility:
16. One possibility: 17. 18.
limx → �
f(x) � �; limx → ��
f(x) � �� limx → �
f(x) � ��; limx → ��
f(x) � �
19. 20. 21.
limx → �
f(x) � ��; limx → ��
f(x) � � limx → �
f(x) � �; limx → ��
f(x) � �� limx → �
f(x) � �; limx → ��
f(x) � �
22. 23. 24.
limx → �
f(x) � �; limx → ��
f(x) � � limx → �
f(x) � �; limx → ��
f(x) � � limx → �
f(x) � ��; limx → ��
f(x) � ��
[–4, 3] by [–20, 90][–3, 5] by [–50, 50][–2, 6] by [–100,
25]
[–5, 5] by [–14, 6][–10, 10] by [–100, 130][–8, 10] by [–120,
100]
[–100, 100] by [–2000, 2000]
[–5, 5] by [–15, 15][–5, 3] by [–8, 3]
[–50, 50] by [–1000, 1000][–50, 50] by [–1000, 1000][–100, 100]
by [–1000, 1000]
x
y
5
40
x
y
5
5
x
y
10
20
ADDITIONAL ANSWERS 925
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 925
-
29. (a) There are 3 zeros: they are �2.5, 1, and 1.1. 30. (b)
There are 3 zeros: they are 0.4, approximately 0.429 �actually
�37��,and 3. 31. (c) There are 3 zeros: approximately �0.273
�actually ��1
31��, �0.25, and 1. 32. (d) There are 3 zeros: �2, 0.5,
and 3. 33. �4 and 2 35. �23
� and ��13
� 37. 0, ��23
�, and 1
39. Degree 3; zeros: x � 0 (mult. 1, graph crosses x-axis), 40.
Degree 4; zeros: x � 0 (mult. 3, graph crosses x-axis),x � 3 (mult.
2, graph is tangent) x � 2 (mult. 1, graph crosses x-axis)
41. Degree 5; zeros: x � 1 (mult. 3, graph crosses x-axis), 42.
Degree 6; zeros: x � 3 (mult. 2, graph is tangent),x � �2 (mult. 2,
graph is tangent) x � �5 (mult. 4, graph is tangent)
43. 44. 45.
�2.43, �0.74, 1.67 �1.73, 0.26, 4.47 �2.47, �1.46, 1.94
46. 47. 48.
�4.53, 2 �4.90, �0.45, 1, 1.35 �1.98, �0.16, 1.25, 2.77,
3.62
53. f(x) � x3 � 5x2 � 18x � 72 55. f(x) � x3 � 4x2 � 3x � 1256.
f(x) � x3 � 3x2 � x � 1 61. It follows from the Intermediate Value
Theorem. 62. It follows from the Intermediate Value Theorem.
[–3, 4] by [–100, 100][–6, 4] by [–100, 20][–5, 3] by [–20,
90]
[–3, 3] by [–10, 10][–2, 5] by [–8, 22][–3, 2] by [–10, 10]
x
y
10
100,000
y
10
–10
x–5 5
x
y
4
5
x
y
6
10
926 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 926
-
63. (a) (c)
65. (a) 66. (a) The height of the box will be x, the width will
be 15 � 2x, and the length 60 � 2x.(b) Any value of x between
approximately 0.550 and 6.786 inches.
67. 0 x � 0.929 or 3.644 � x 569. True. Because f is continuous
and f(1) � �2 and f(2) � 2, the Intermediate Value Theorem assures
us
that the graph of f crosses the x-axis between x � 1 and x �
2.70. False. If a � 0, the graph of g is obtained by translating
the graph of f a units to the left.77. The exact behavior near x �
1 is hard to see. A zoomed-in view around the point (1, 0) suggests
that
the graph just touches the x-axis at 0 without actually crossing
it — that is, (1, 0) is a local maximum.One possible window is
[0.9999, 1.0001] by [�1 � 10�7, 1 � 10�7].
78. This also has a maximum near x � 1 — but this time a window
such as [0.6, 1.4] by [�0.1, 0.1] reveals that the graph actually
rises above the x-axis and has a maximum at (0.999, 0.025). 79. A
maximum and minimum are not visible in the standard window, but
canbe seen on the window [0.2, 0.4] by [5.29, 5.3]. 80. A maximum
and minimum are not visible in the standard window, but can be seen
on thewindow [0.95, 1.05] by [�6.0005, �5.9995].81. The graph of y
� 3(x3 � x) increases, then decreases, then increases; the graph of
y � x3 only increases. Therefore, this graph can not beobtained
from the graph of y � x3 by the transformations studied in Chapter
1 (translations, reflections, and stretching/shrinking). Since the
rightside includes only these transformations, there can be no
solution. 82. The graph of y � x4 has a “flat bottom,” while the
graph of y � x4 �3x3 � 2x � 3 is “bumpy.” Therefore, this graph
cannot be obtained from the graph of y � x4 through the
transformations of Chapter 1. Since theright side includes only
these transformations, there can be no solution.
(b)
84. (a) Note that f(a) � an and f(�a) � �an; (b) y � an/(n � 1)
� an � 1(x � a1/(n � 1))
m � �yx
2
2
�
�
yx
1
1� � �
�
�
aa
n �
�
aa
n
� � ��
�
22aa
n
� � an � 1 (c) y � 9x � 6�3�, y � x3
SECTION 2.4 Quick Review 2.4
7. 4(x � 5)(x � 3) 8. x(3x � 2)(5x � 4)
Exercises 2.4
1. f(x) � (x � 1)2 � 2; �xf�
(x)1
� � x � 1 � �x �
21
� 2. f(x) � (x2 � x � 1)(x � 1) � 2; �xf�
(x)1
� � x2 � x � 1 � �x �
21
�
[–5, 5] by [–30, 30]
(c) The line L also crosses the graph off(x) at (�2, �13).
[1.8, 2.2] by [6, 8]
83. (a) Substituting x � 2, y � 7, we find that7 � 5(2 � 2) � 7,
so Q is on line L,and also f(2) � �8 � 8 � 18 � 11 � 7,so Q is on
the graph of f(x).
[0, 0.8] by [0, 1.20]
[0, 60] by [–10, 210][0, 60] by [–10, 210]
ADDITIONAL ANSWERS 927
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 927
-
3. f(x) � (x2 � x � 4)(x � 3) � 21; �xf�
(x)3
� � x2 � x � 4 � �x
2�
13
� 4. f(x) � �2x2 � 5x � �72��(2x � 1) � �92
�;
�2x
f(�
x)1
� � 2x2 � 5x � �72
� � �2x
9�
/21
� 5. f(x) � (x2 � 4x � 12)(x2 � 2x � 1) � 32x � 18;
�x2 �
f(2xx)� 1
� � x2 � 4x � 12 � �x�2
3�
2x2x
�
�
181
� 6. f(x) � (x2 � 3x � 5)(x2 � 1); �x2
f(�
x)1
� � x2 � 3x � 5
33. ��1, �2
�
,1�3, �6�; 1 34. �
�1, ��
21,,�
�
73, �14
�; �73
� 35. ��1
�
,1�
,3�
,2�9
�; �32
� 36. ; ��43
� and �32
�
49. Rational zero: �32
�; irrational zeros: ��2�
53. Rational: �1 and 4; irrational: ��2� 54. Rational: �1 and 2;
irrational: ��5� 55. Rational: ��12
� and 4; irrational: none
56. Rational: �23
�; irrational: about �0.6823 61. (c) (x � 2)(x3 � 4x2 � 3x � 19)
(d) One irrational zero is x � 2.04.(e) f(x) � (x � 2)(x � 2.04)(x2
� 6.04x � 9.3116)
62. (a) D � 0.0669t3 � 0.7420t2 � 2.1759t � 0.8250
63. False. (x � 2) is a factor if and only if f(�2) � 0. 64.
True because the remainder is f(1) which is equal to 3.69. (d) x �
0.6527 m 71. (a) Shown is one possible view, on the window [0, 600]
by [0, 500].
(c) P � 0 when t � 523.22 — about 523 days after release.
SECTION 2.5 Exploration 11. f(2i) � (2i)2�i(2i) � 2 � �4 � 2 � 2
� 0; f (�i) � (�i)2 � i(�i) � 2 � �1 � 1 � 2 � 0; no.2. g(i) � i2 �
i � (1 � i) � �1 � i � 1 � i � 0; g(1 � i) � (1 � i)2 � (1 � i) �
(l � i) � � 2i � 2i � 0; no.3. The Complex Conjugate Zeros Theorem
does not necessarily hold true for a polynomial function with
complex coefficients.
Quick Review 2.5
7. �52
� � ��
219��i 8. ��
34
� � ��
447��i
Exercises 2.52. x3 � 2x2 � 3x � 6; zeros: �2, ��3�i;
x-intercept: x � �23. x4 � 2x3 � 5x2 � 8x � 4; zeros: 1 (mult. 2),
�2i; x-intercept: x � 1 4. x4 � 3x3 � 4x2 � 2x; zeros: 0, 1, 1 � i;
x-intercepts: x � 0, x � 1 7. x3 � x2 � 9x � 9 8. x3 � 2x2 � 6x � 8
9. x4 � 5x3 � 7x2 � 5x � 610. x4 � 3x3 � 2x2 � 2x � 4 11. x3 � 11x2
� 43x � 65 13. x5 � 4x4 � x3 � 10x2 � 4x � 815. x4 � 10x3 � 38x2 �
64x � 40 16. x4 � 6x3 � 14x2 � 14x � 5
[0, 600] by [0, 500]
[–1, 8.25] by [0, 5]
�1, �2, �3, �4, �6, �12���
�1, �2, �3, �6
928 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 928
-
27. Zeros: x � 1, x � ��12
� � ��
219��i; f(x) � �
14
�(x � 1)(2x � 1 � �19�i)(2x � 1 � �19�i)
28. Zeros: x � 3, x � �72
� � ��
243��i; f(x) � �
14
�(x � 3)(2x � 7 � �43�i); (2x � 7 � �43�i)
29. Zeros: x � �1, x � ��12
� � ��
223��i; f(x) � �
14
�(x � 1)(x � 1)(2x � 1 � �23�i)(2x � 1 � �23�i)
30. Zeros: x � �2, x � �13
�, x � ��12
� � ��
23��i; f(x) � �
14
�(x � 2)(3x � 1)(2x � 1 � �3�i)(2x � 1 � �3�i)
31. Zeros: x � ��73
�, x � �32
�, x � 1 � 2i; f(x) � (3x � 7)(2x � 3)(x � 1 � 2i)(x � 1 �
2i)
32. Zeros: x � ��35
�, x � 5, x � �32
� � i; f(x) � (5x � 3)(x � 5)(2x � 3 � 2i)(2x � 3 � 2i)
33. Zeros: x � ��3�, x � 1 � i; f(x) � (x � �3� )(x � �3� )(x �
1 � i)(x � 1 � i)34. Zeros: x � ��3�, x � �4i; f(x) � (x � �3� )(x
� �3� )(x � 4i)(x � 4i)35. Zeros: x � ��2�, x � 3 � 2i; f(x) � (x �
�2�)(x � �2�)(x � 3 � 2i)(x � 3 � 2i)36. Zeros: x � ��5�, x � 1 �
3i; f(x) � (x � �5�)(x � �5� )(x � 1 � 3i)(x � 1 � 3i)49. f(x) �
�2x4 � 12x3 � 20x2 � 4x � 30 50. f(x) � 2x4 � 8x3 � 22x2 � 28x �
20
51. (a) D � �0.0820t3 � 0.9162t2 � 2.5126t � 3.3779
(b) Sally walks toward the detector, turns and walks away (or
walks backward), then walks toward the detector again.
(c) t � 1.81 sec (D � 1.35 m) and t � 5.64 sec (D � 3.65 m).
52. (a) D � 0.2434t2 � 1.7159t � 4.4241
(b) Jacob walks toward the detector, then turns and walks away
(or walks backward).
(c) The model “changes direction” at t � 3.52(when D � 1.40
m).
53. False. If 1 � 2i is a zero, then 1 � 2i must also be a
zero.54. False. The polynomial f(x) � x(x � 1)(x � 2) � x3 � 3x2 �
2x has degree 3, real coefficients, and no non-real zeros.
[–1, 9] by [0, 6]
[–1, 9] by [0, 5]
ADDITIONAL ANSWERS 929
5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 929
-
SECTION 2.6 Exploration 1
1. g(x) � �x �
12
� 2. h(x) � ��x �
15
� 3. k(x) � �x �
34
� � 2
Exercises 2.61. Domain: all x �3; lim
x → �3�f(x) � ��, lim
x → �3�f(x) � � 2. Domain: all x 1; lim
x → 1�f(x) � �; lim
x → 1�f(x) � ��
3. Domain: all x �2, 2; limx → �2�
f(x) � ��, limx → �2�
f(x) � �, limx → 2�
f(x) � �, limx → 2�
f(x) � ��
4. Domain: all x �1, 1; limx → �1�
f(x) � �, limx → �1�
f(x) � ��, limx → 1�
f(x) � ��, limx → 1�
f(x) � �
5. Translate right 3 units. 6. Translate left 5 units,
vertically 7. Translate left 3 units, reflect across
x-axis,Asymptotes: x � 3, y � 0 stretch by 2, reflect across
x-axis. vertically stretch by 7, translate up 2 units.
Asymptotes: x � �5, y � 0 Asymptotes: x � �3, y � 2
8. Translate right 1 unit, translate up 9. Translate left 4
units, vertically 10. Translate right 5 units, vertically 3 units.
Asymptotes: x � 1, y � 3 stretch by 13, translate down 2 units.
stretch by 11, reflect across x-axis,
Asymptotes: x � �4, y � �2 translate down 3 units. Asymptotes: x
� 5, y � �3
19. Vertical asymptote: none; Horizontal asymptote: y � 2; limx
→ ��
f(x) � limx → �
f(x) � 2
20. Vertical asymptote: none; Horizontal asymptote: y � 3; limx
→ ��
f(x) � limx → �
f(x) � 3
21. Vertical asymptotes: x � 0, x � 1; Horizontal asymptote: y �
0; limx → 0�
f(x) � �, limx → 0�
f(x) � ��, limx → 1�
f(x) � ��,lim
x → 1�f(x) � �, lim
x → ��f(x) � lim
x → �f(x) � 0
22. Vertical asymptotes: x � �3, x � 0; Horizontal asymptote: y
� 0; limx → �3�
f(x) � ��, limx → �3�
f(x) � �, limx → 0�
f(x) � �,lim
x → 0�f(x) � ��, lim
x → ��f(x) � lim
x → �f(x) � 0
x
y
10
20
x
y
6
8
x
y
6
6
x
y
6
10
x
y
4
5
x
y
5
5
[–8, 2] by [–5, 5][–1, 9] by [–5, 5][–3, 7] by [–5, 5]
930 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 930
-
23. Intercepts: �0, �23�� and (2, 0) 24. Intercepts: �0,
��23
�� and (�2, 0) 25. No intercepts Asymptotes: x � �1, x � 3,
Asymptotes: x � �3, x � 1, Asymptotes: x � �1, x � 0,and y � 0 and
y � 0 x � 1, and y � 0
26. No intercepts 27. Intercepts: (0, 2), (�1.28, 0), and 28.
Intercepts: (0, �3), (�1.84, 0), and Asymptotes: x � �2, x � 0,
(0.78, 0); Asymptotes: x � 1, (2.17, 0); Asymptotes: x � �2,x � 2,
and y � 0. x � �1, and y � 2 x � 2, and y � �3
29. Intercept: �0, �32�� 30. Intercepts: �0, ��73��, (�1.54, 0),
and (4.54, 0) Asymptotes: x � �2, y � x � 4 Asymptotes: x � �3, y �
x � 6
31. (d); Xmin � �2, Xmax � 8, Xscl � 1, and Ymin � �3, Ymax � 3,
Yscl � 132. (b); Xmin � �6, Xmax � 2, Xscl � 1, and Ymin � �3, Ymax
� 3, Yscl � 133. (a); Xmin � �3, Xmax � 5, Xscl � 1, and Ymin � �5,
Ymax � 10, Yscl � 134. (f); Xmin � �6, Xmax � 2, Xscl � 1, and Ymin
� �5, Ymax � 5, Yscl � 135. (e); Xmin � �2, Xmax � 8, Xscl � 1, and
Ymin � �3, Ymax � 3, Yscl � 136. (c); Xmin � �3, Xmax � 5, Xscl �
1, and Ymin � �3, Ymax � 8, Yscl � 1
37. Intercept: �0, ��23��; asymptotes: x � �1, x � �32
�, y � 0; limx → �1�
f(x) � �, limx → �1�
f(x) � ��, limx → (3/2)�
f(x) � ��,
limx → (3/2)�
f(x) � �; Domain: x �1, �32
�; Range: ���, ��1265�� � (0, �); Continuity: all x �1,
�32�;
Increasing: (��, �1), ��1, �14��, Decreasing: ��14
�, �32
��, ��32�, ��; Unbounded; Local Maximum at ��14�, ��
1265��; Horizontal asymptote: y � 0;
Vertical asymptotes: x � �1, x � �32
�; End behavior: limx → ��
f(x) � limx → �
f(x) � 0[–4.7, 4.7] by [–3.1, 3.1]
[–30, 30] by [–40, 20][–20, 20] by [–20, 20]
[–5, 5] by [–8, 2][–5, 5] by [–4, 6][–4, 4] by [–5, 5]
[–4.7, 4.7] by [–10, 10][–6, 4] by [–5, 5][–4, 6] by [–5, 5]
ADDITIONAL ANSWERS 931
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 931
-
38. Intercept: �0, �23��; asymptotes: x � �3, x � �1, y � 0;
limx → �3� g(x) � �, limx → �3� g(x) � ��, limx → �1� g(x) �
��,lim
x → �1�g(x) � � Domain: x �3, �1; Range: (��, �2] � (0, �);
Continuity: all x �3, �1; Increasing: (��, �3), (�3, �2];
Decreasing: [�2, �1), (�1, �); No symmetry; Unbounded; Local
maximum at (�2, �2); Horizontal asymptote: y � 0; Vertical
asymptotes: x � �3, x � �1; End behavior: lim
x → ��g(x) � lim
x → �g(x) � 0
39. Intercepts: �0, �112��, (1, 0); asymptotes: x � �3, x � 4, y
� 0; limx → �3� h(x) � ��, limx → �3� h(x) � �, limx → 4� h(x) �
��,
limx → 4�
h(x) � � Domain: x �3, 4; Range: (��, �); Continuity: all x �3,
4; Decreasing: (��, �3), (�3, 4), (4, �); No symmetry; Unbounded;
No extrema; Horizontal asymptote: y � 0; Vertical asymptotes: x �
�3, x � 4; End behavior: lim
x → ��h(x) � lim
x → �h(x) � 0
40. Intercepts: (�1, 0), (0, �0.1); asymptotes: x � �2, x � 5, y
� 0; limx → �2�
k(x) � ��, limx → �2�
k(x) � �, limx → 5�
k(x) � ��,
limx → 5�
k(x) � � Domain: x �2, 5; Range: (��, �); Continuity: all x �2,
5; Decreasing: (��, �2), (�2, 5), (5, �); No symmetry; Unbounded;
No extrema; Horizontal asymptote: y � 0; Vertical asymptotes: x �
�2, x � 5; End behavior: lim
x → ��k(x) � lim
x → �k(x) � 0
41. Intercepts: (�2, 0), (1, 0), �0, �29��; asymptotes: x � �3,
x � 3, y � 1; limx → �3� f(x) � �, limx → �3� f(x) � ��, limx → 3�
f(x) � ��,lim
x → 3�f(x) � � Domain: x �3, 3; Range: (��, 0.260] � (1, �);
Continuity: all x �3, 3;
Increasing: (��, �3), (�3, �0.675); Decreasing: (�0.675, 3), (3,
�); No symmetry; Unbounded; Local maximum at (�0.675, 0.260);
Horizontal asymptote: y � 1; Vertical asymptotes: x � �3, x � 3;
End behavior: lim
x → ��f(x) � lim
x → �f(x) � 1
42. Intercepts: (�1, 0), (2, 0), �0, �14��; asymptotes: x � �2,
x � 4, y � 1; limx → �2� g(x) � �, limx → �2� g(x) � ��, limx → 4�
f(x) � ��,lim
x → 4�g(x) � � Domain: x �2, 4; Range: (��, 0.260] � (1, �);
Continuity: all x �2, 4; Increasing (��, �2), (�2, 0.324];
Decreasing: [0.324, 4), (4, �); No symmetry; Unbounded; Local
maximum at (0.324, 0.260); Horizontal asymptote: y � 1; Vertical
asymptotes: x � �2, x � 4; End behavior: lim
x → ��g(x) � lim
x → �g(x) � 1
[–9.4, 9.4] by [–3, 3]
[–9.4, 9.4] by [–3, 3]
[–9.4, 9.4] by [–1, 1]
[–5.875, 5.875] by [–3.1, 3.1]
[–6.7, 2.7] by [–5, 5]
932 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 932
-
43. Intercepts: (�3, 0), (1, 0), �0, ��32��; asymptotes: x � �2,
y � x; limx → �2� h(x) � �, limx → �2� h(x) � ��Domain: x �2,
Range: (��, �); Continuity: all x �2; Increasing: (��, �2), (�2,
�); No symmetry; Unbounded; No extrema; Horizontal asymptote: none;
Vertical asymptote: x � �2; Slant asymptote: y � x; End behavior:
lim
x → ��h(x) � ��, lim
x → �h(x) � �
44. Intercepts: (�1, 0), (2, 0), �0, �23��; asymptotes: x � 3, y
� x � 2; limx → 3� k(x) � ��, limx → 3� k(x) � �Domain: x 3; Range:
(��, 1] � [9, �); Continuity: all x 3; Increasing: (��, 1], [5, �);
Decreasing: [1, 3), (3, 5]; No symmetry; Unbounded; Local max at
(1, 1), local min at (5, 9); Horizontal asymptote: none; Vertical
asymptote: x � 3; Slant asymptote: y � x � 2; End behavior: lim
x → ��k(x) � ��, lim
x → �k(x) � �
45. y � x � 3 46. y � 2x � 4 47. y � x2 � 3x � 6(a) (a) (a)
(b) (b) (b)
48. y � x2 � x � 1 49. y � x3 � 2x2 � 4x � 6(a) (a)
(b) (b)
[–20, 20] by [–5000, 5000][–50, 50] by [–1500, 2500]
[–5, 5] by [–100, 200][–8, 8] by [–20, 40]
[–50, 50] by [–1500, 2500][–40, 40] by [–40, 40][–40, 40] by
[–40, 40]
[–10, 10] by [–30, 60][–15, 10] by [–30, 20][–10, 20] by [–10,
30]
[–9.4, 9.4] by [–10, 20]
[–9.4, 9.4] by [–15, 15]
ADDITIONAL ANSWERS 933
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 933
-
50. y � x3 � x(a) There are no vertical asymptotes.(b)
51. Intercept: �0, �45��; 52. Intercepts: ���12
�, 0�, (0, 0); Domain: (��, �); Range: [0.773, 14.227]; Domain:
(��, �); Range: [�0.028, 9.028]; Continuity: (��, �); Increasing:
[�0.245, 2.445]; Continuity: (��, �); Increasing: [�0.235, 3.790];
Decreasing: (��, �0.245], [2.445, �); Decreasing: (��, �0.235],
[3.790, �); No symmetry; Bounded; No symmetry; Bounded; Local max
at (2.445, 14.227), local min at (�0.245, 0.773); Local max at
(3.790, 9.028), local min at (�0.235, �0.028); Horizontal
asymptote: y � 3; Vertical asymptote: none; Horizontal asymptote: y
� 4; Vertical asymptote: none; End behavior: lim
x → ��f(x) � lim
x → �f(x) � 3 End behavior: lim
x → ��g(x) � lim
x → �g(x) � 4
53. Intercepts: (1, 0), �0, �12��; 54. Intercepts: (�3 2�, 0),
(0, �1); Domain: x 2; Range: (��, �); Continuity: x 2; Domain: x
�2; Range: (��, �); Continuity: x �2; Increasing: [�0.384, 0.442],
[2.942, �); Increasing: [�3.104, �2], (�2, �); Decreasing: (��,
�0.384], [0.442, 2), (2, 2.942]; Decreasing: (��, �3.104]; No
symmetry; Not bounded; No symmetry; Not bounded; Local max at
(0.442, 0.586), local min at (�0.384, 0.443) Local min at (�3.104,
28.901); and (2.942, 25.970); Horizontal asymptote: none;
Horizontal asymptote: none; Vertical asymptote: x � 2; Vertical
asymptotes: x � �2; End behavior: lim
x → ��h(x) � lim
x → �h(x) � �; End behavior: lim
x → ��k(x) � lim
x → �k(x) � �;
End-behavior asymptote: y � x2 � 2x � 4 End-behavior asymptote:
y � x2 � 2x � 4
[–10, 10] by [–20, 50][–10, 10] by [–20, 50]
[–10, 15] by [–5, 10][–15, 15] by [–5, 15]
[–20, 20] by [–5000, 5000]
934 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 934
-
55. Intercepts: (1.755, 0), (0, 1); 56. Intercepts: (�1.189, 0),
(0, �2.5);
Domain: x �12
�; Range: (��, �); Continuity: x �12
�; Domain: x 2; Range: (��, �); Continuity: x 2;
Increasing: [�0.184, �12�), ��12�, ��; Increasing: [2.899, �);
Decreasing: (��, �0.184]; Decreasing: (��, 2), (2, 2.899]; No
symmetry; Not bounded; No symmetry; Not bounded; Local min at
(�0.184, 0.920); Local min at (2.899, 37.842); Horizontal
asymptote: none; Vertical asymptote: x � �
12
�; Horizontal asymptote: none; Vertical asymptote: x � 2;
End behavior: limx → ��
f(x) � limx → �
f(x) � �; End behavior: limx → ��
g(x) � limx → �
g(x) � �;
End-behavior asymptote: y � �12
�x2 � �34
�x � �18
� End-behavior asymptote: y � 2x2 � 2x � 3
57. Intercept: (0, 1); 58. Intercepts: (�1.108, 0), (0, �1); 59.
Intercepts: (1, 0), �0, ��12��; Asymptote: x � �1; Asymptotes: x �
�1; Asymptote: x � �2; End-behavior asymptote: End-behavior
asymptote: End-behavior asymptote:y � x3 � x2 � x � 1 y � 2x3 � 2x
� 1 y � x4 � 2x3 � 4x2 � 8x � 16
60. Intercepts: (�1, 0), (0, �1); 61. Intercepts: (�1.476, 0),
(0, �2); 62. Intercepts: (1, 0), (0, �4); Asymptote: x � 1;
Asymptote: x � 1; Asymptote: x � �1; End-behavior asymptote:
End-behavior asymptote: y � 2 End-behavior asymptote: y � 3y � x4 �
x3 � x2 � x � 1
63. False. �x2 �
11
� is a rational function and has no vertical asymptotes.
64. False. A rational function is the quotient of two
polynomials, and �x2 � 4� is not a polynomial.69. (a) No: the
domain of f is (��, 3) � (3, �); the domain of g is all real
numbers.(b) No: while it is not defined at 3, it does not tend
toward �� on either side.(c) Most grapher viewing windows do not
reveal that f is undefined at 3. (d) Almost—but not quite, they are
equal for all x 3.70. (a) The functions are identical at all points
except x � 1 (b) The functions are identical except at x � �1.(c)
The functions are identical except at x � �1. (d) The functions are
identical except at x � 1.
[–5, 5] by [–25, 50]
[–5, 5] by [–10, 10][–5, 5] by [–5, 5]
[–10, 10] by [–200, 400][–3, 3] by [–20, 40][–5, 5] by [–30,
30]
[–10, 10] by [–20, 60][–5, 5] by [–10, 10]
ADDITIONAL ANSWERS 935
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 935
-
71. (b) If f(x) � kxa, where a is a negative integer, 72. (a)
(c)then the power function f is also a rational function.
73. Horizontal asymptotes: y � �2 and y � 2; 74. Horizontal
asymptotes: y � �3;
Intercepts: �0, ��32��, ��32
�, 0�; Intercepts: �0, �53��, ���53
�, 0�;
h(x) �� h(x) ��
75. Horizontal asymptotes: y � �3; 76. Horizontal asymptotes: y
� �2;
Intercepts: �0, �54��, ��53
�, 0�; Intercepts: (0, 2), (1, 0);
f(x) �� f(x) ��
SECTION 2.7 Quick Review 2.71. 2x2 � 8x 2. x2 � 2x � 1 3. LCD:
36; ��
316� 4. LCD: x(x � 1); �
2xx2 �
�
x1
�
6. LCD: (x � 2)(x � 3)(x � 2);�(x �
�22x)(
�
x �8
2)� if x 3 7. �
3 �4�17�� 8. �
5 �4�33�� 9. �
�1 �3
�7�� 10. �
3 �23�5��
Exercises 2.71. x � �1 4. x � �
11 �8�73�� � 2.443 or x � �
11 �8�73�� � 0.307
[–7, 13] by [–3, 3][–10, 10] by [–5, 5]
�2x�
�
21x
� x � 0
2 x 0
�5x�
�
34x
� x � 0
��
5x�
�
3x4
� x 0
[–5, 5] by [–5, 5][–5, 5] by [–5, 5]
�3xx�
�
35
� x � 0
��
3xx
�
�
53
� x 0
�2xx�
�
23
� x � 0
��
2xx
�
�
32
� x 0
[0.5, 3.5] by [0, 7][0.5, 3.5] by [0, 7]
936 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 936
-
5. x � �4 or x � 3, the latter is extraneous. 6. x � �13 �
1�6
105�� � 1.453 or x � �
13 �1�6
105�� � 0.172
11. x � �12
� or x � �1, the latter is extraneous.
12. x � �32
� or x � �4, the latter is extraneous. 13. x � ��13
� or x � 2, the latter is extraneous.
14. x � ��34
� or x � 1, the latter is extraneous.
23. x � 3 � �2� � 4.414 or x � 3 � �2� � 1.586 24. x � ��3 �
2�31�� � 1.284 or x � �
�3 �2�31�� � �4.284
26. x � �1 �
6�13�� � 0.768 or x � �
1 �6�13�� � �0.434
31. (a) The total amount of solution is (125 � x) mL; of this,
the amount of acid is x plus 60% of the original amount, or x �
0.6(125).
(c) C(x) � �xx�
�
17255
� � 0.83; x � 169.12 mL 32. (a) C(x) � �x �
x0�
.3150(1000)
� � �xx�
�
13050
�
33. (a) C(x) � �3000 �
x2.12x� (c) 6350 hats per week
34. (c) limt→�
P(t) � limt→� �500 � �t9�00200�� � 500, so the bear population
will never exceed 500.
35. (a) P(x) � 2x � �36x4
� (b) x � 13.49 (a square); P � 53.96
36. (a) A(x) � (x � 1.75)��4x0� � 2.5� (b) x � 5.29, so the
dimensions are about 7.04 in. � 10.06 in.; A � 70.8325 in2
37. (a) S � �2�x3 �
x1000� (b) Either x � 1.12 cm and h � 126.88 cm or x � 11.37 cm
and h � 1.23 cm
38. (a) A(x) � (x � 4)��10x00� � 4� (b) x � �1000� � 31.62, so
the dimensions are about 35.62 ft � 35.62 ft; A � 1268.98 ft2
40. (a) P(x) � 2x � �40
x0
� (b) 7.1922 m � 27.8078 m
41. (a) D(t) � �4.
47.575
�
tt
� 42. (a) T � �1x7� � �
x �53
43�
43. (a) 44. (a) 45. False. An extraneous solution is a solution
of the equation cleared of fractions that is not a solution of the
original equation.
46. True. For a fraction to be equal to zero,the numerator must
be zero and 1 is not zero.
(b) About 2102 wineries
51. (a) f(x) � �xx
2
2�
�
22xx
� (c) f(x) � {(d) The graph appears to be the horizontal
line
y � 1 with holes at x � �2 and x � 0.
[–4.7, 4.7] by [–3.1, 3.1]
x �2, 0x � �2 or x � 0
1,undefined,
[0, 35] by [0, 3000][0, 15] by [0, 120]
ADDITIONAL ANSWERS 937
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 937
-
SECTION 2.8 Exploration 1
1. (a) 2. (a)
(b) (b)
3. (a) (b)
Quick Review 2.8
7. �2xx
2
2�
�
75xx�
�
23
� 9. (a) �1, ��12
�, �3, ��32
� (b) (x � 1)(2x � 3)(x � 1)
10. (a) �1, ��13
�, �2, ��23
�, �4, ��43
�, �8, ��83
� (b) (x � 2)(x � 1)(3x � 4)
Exercises 2.81. (a) x � �2, �1, 5 (b) �2 x �1 or x � 5 (c) x �2
or �1 x 5 2. (a) x � 7, ��
13
�, �4
(b) �4 x ��13
� or x � 7 (c) x �4 or ��13
� x 7 3. (a) x � �7, �4, 6 (b) x �7 or �4 x 6 or x � 6
(c) �7 x �4 4. (a) x � ��35
�, 1 (b) x ��35
� or x � 1 (c) ��35
� x 1 5. (a) x � 8, �1 (b) �1 x 8 or x � 8
(c) x �1 6. (a) x � �2, 9 (b) �2 x 9 or x � 9 (c) x �2 21. (a)
(��, �) (b) (��, �)(c) There are no solutions. (d) There are no
solutions. 22. (a) There are no solutions. (b) There are no
solutions. (c) (��, �)
(d) (��, �) 23. (a) x �43
� (b) (��, �) (c) There are no solutions. (d) x � �43
� 24. (a) x �32
� (b) (��, �)
(c) There are no solutions. (d) x � �32
� 25. (a) x � 1 (b) x � ��32
�, 4 (c) ��32
� x 1 or x � 4 (d) x ��32
�, or 1 x 4
26. (a) x � �72
�, �1 (b) x � �5 (c) �5 x �1 or x � �72
� (d) x �5 or �1 x �72
� 27. (a) x � 0, �3 (b) x �3
(c) x � 0 (d) �3 x 0 28. (a) x � 0, ��92
� (b) None (c) x ��92
�, 0 (d) None 29. (a) x � �5
(b) x � ��12
�, x � 1, x �5 (c) �5 x ��12
� or x � 1 (d) ��12
� x 1 30. (a) x � 1 (b) x � 4, x � �2
(c) �2 x 1 or x � 4 (d) 1 x 4 31. (a) x � 3 (b) x � 4, x 3 (c) 3
x 4 or x � 4 (d) f(x) is never negative.
32. (a) None (b) x � 5 (c) 5 x � (d) None 33. (��, �2) � (1, 2)
37. (��, �4) � (3, �) 39. [�1, 0] � [1, �)41. (0, 2) � (2, �)
43. ��4, �12�� 47. (��, 0) � (�3 2�, �)49. (��, �1) � [1, 3) 50.
(��, �5) � (�2, 1) 51. [�3, �)
[–5, 5] by [–3000, 2000]
Positive Negative Negative(+)(+) (–)(–) (+)(+) (+)(–) (+)(+)
(+)(–)
2–4x
[–3, 1] by [–30, 20][–5, 5] by [–250, 50]
Positive Positive Negative(+)(–) (–)(+) (+)(–) (–)(+) (+)(–)
(+)(+)
–1–2xNegative Negative Positive
(+)(–)(+) (+)(–)(+) (+)(+)(+)
2–3
x
938 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 938
-
59. 0 in. � x � 0.69 in. or 4.20 in. � x � 6 in.61. (b) 1.12 cm
� x � 11.37 cm, 1.23 cm � h � 126.88 cm 62. (a) R � �
x2�
.32x.3
�
63. (a) y � 993.870x � 19025.768 64. (a) y � 7.883x3 � 214.972x2
� 6479.797x � 62862.27865. False, because the factor x4 does not
change sign at x � 0. 66. True, because the factor (x � 2) changes
sign at x � �2.
71. Vertical asymptotes: x � �1, x � 3; x-intercepts: (�2, 0),
(1, 0); y-intercept: �0, �43��
By hand: Grapher
72. Vertical asymptotes: x � �4, x � 0; x-intercept: (3, 0);
y-intercept: none
Sketch: Graph:
73. (a) x � 3 13 ⇒ 3x � 9 1 ⇒ 3x � 5 � 4 1 ⇒ f(x) � 4 1.(b) If x
stays within the dashed vertical lines, f(x) will stay within the
dashed horizontal lines.(c) x � 3 0.01 ⇒ 3x � 9 0.03 ⇒ 3x � 5 � 4
0.03 ⇒ f(x) � 4 0.03. The dashed lines would be closer when x � 3
and
y � 4.74. When x2 � 4 � 0, y � 1, and when x2 � 4 � 0, y �
0.
[–10, 10] by [–10, 10] [–20, 0] by [–1000, 1000]x
y
10
600
Positive Negative Positive Positive30–4
0unde
fine
d
unde
fine
d
x
(–)4
(–)(–)
(–)4
(–)(+)
(–)4
(+)(+)
(+)4
(+)(+)
[–5, 5] by [–5, 5] [0, 10] by [–40, 40]
x
y
105
30
–10
-30
Negative3–1 1–2
0 unde
fine
d
0 unde
fine
dx
(–)(–)2
(–)(–)
Negative
(–)(+)2
(–)(–)
Positive
(–)(+)2
(–)(+)
Negative
(+)(+)2
(–)(+)
Positive
(+)(+)2
(+)(+)
ADDITIONAL ANSWERS 939
support:
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 939
-
CHAPTER 2 REVIEW EXERCISES1. y � �x � 5 2. y � �2x
3. Starting from y � x2, translate 4. Starting from y � x2,
translate 5. Vertex: (�3, 5); axis: x � �3right 2 units and
vertically stretch left 3 units and reflect across the 6. Vertex:
(5, �7); axis: x � 5by 3 (either order), then translate x-axis
(either order), then translate 7. Vertex: (�4, 1); axis: x � �4up 4
units. up 1 unit. 8. Vertex: (1, �1); axis: x � 1
11. y � �12
�(x � 3)2 � 2
12. y � ��12
�(x � 4)2 � 5
13. 14. 15. 16.
19. The force F needed varies directly with the distance x from
its resting position, with constant of variation k.
20. The area of a circle A varies directly with the square of
its radius. 21. k � 4, a � �13
�, f is increasing in the first quadrant, f is odd.
22. k � �2, a � �34
�, f is decreasing in the fourth quadrant, f is not defined for
x 0. 23. k � �2, a � �3, f is
increasing in the fourth quadrant, f is odd. 24. k � �23
�, a � �4, f is decreasing in the first quadrant, f is even.
25. 2x2 � x � 1 � �x �
23
� 26. x3 � x2 � x � 1 � �x �
52
� 27. 2x2 � 3x � 1 � ��
x22x�
�
43
� 28. x3 � 2x2 � 1 � �3x
�
�
71
�
37. �1, �2, �3, �6, ��12
�, ��32
� ; ��32
� and 2 are zeros.
38. �1, �7, ��12
�, ��72
�, ��13
�, ��73
�, ��16
�, ��76
�; �73
� is a zero.
49. Rational: 0. Irrational: 5 � �2�. No nonreal zeros. 50.
Rational: �2. Irrational: ��3�. No nonreal zeros.51. Rational:
none. Irrational: approximately �2.34, 0.57, 3.77. No nonreal
zeros.
52. Rational: none. Irrational: approximately �3.97, �0.19. Two
nonreal zeros. 53. ��32
�, 3 � i;
f(x) � (2x � 3)(x � 3 � i)(x � 3 � i) 54. �45
�, 2 � �7�; f(x) � (5x � 4)(x � 2 � �7�)(x � 2 � �7�)
55. 1, �1, �23
�, and ��52
�; f(x) � (3x � 2)(2x � 5)(x � 1)(x � 1)
56. 3 � i, 1 � 2i; f(x) � (x � 1 � 2i)(x � 1 � 2i)(x � 3 � i)(x
� 3 � i)
[–6, 7] by [–50, 30][–4, 3] by [–30, 30][–2, 4] by [–50,
10][–10, 7] by [–50, 10]
x
y
10
10
x
y
6
10
[–5, 5] by [–5, 5][–15, 5] by [–15, 5]
940 ADDITIONAL ANSWERS
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 940
-
59. f(x) � (2x � 3)(x � 1)(x2 � 2x � 5)60. f(x) � (3x � 2)(x �
1)(x2 � 4x � 5) 63. 6x4 � 5x3 � 38x2 � 5x � 6 65. x4 � 4x3 � 12x2 �
32x � 6467. Translate right 5 units and vertically stretch by 2
(either order), then translate down 1 unit.; Horizontal asymptote:
y � �1; vertical asymptote: x � 5. 68. Translate left 2 units and
reflect across x-axis (either order),then translate up 3 units.;
Horizontal asymptote: y � 3; vertical asymptote: x � �2.
69. Asymptotes: y � 1, x � �1, and 70. Asymptotes: y � 2, x �
�3, and 71. End behavior asymptote: y � x � 7; x � 1. Intercept:
(0, �1). x � 2. Intercept: �0, ��76��. Vertical asymptote: x �
�3.
Intercept: �0, �53��.
72. End behavior asymptote: y � x � 6; 73. y-intercept: �0,
�52��, x-intercept: (�2.55, 0); Vertical asymptote: x � �3. Domain:
x �2; Range: (��, �); Continuity: all x �2;
Intercepts: approx. (�1.54, 0), (4.54, 0) and �0, ��73��.
Decreasing: (��, �2), (�2, 0.82]; Increasing: [0.82, �); Unbounded;
Local minimum: (0.82, 1.63); Vertical asymptote: x � �2;
End-behavior asymptote: y � x2 � x;
limx → ��
f(x) � limx → �
f(x) � �
74. y-intercept: (0, �1), x-intercepts: (�1.27, 0), (1.27, 0);
79. [�3, �2) � (2, �)Domain: x 1; Range: (��, �); 80. (��, �2) �
(1, 3)Continuity: all x 1; Decreasing: (��, 1), (1, �); Unbounded;
No extrema; Vertical asymptote: x � 1; 81. x � �3, x � �
12
�
End-behavior asymptote: y � �x3 � x2; 82. (1, 4) � (4, �)lim
x → ��f(x) � �; lim
x → � f(x) � ��
[–4.7, 4.7] by [–10, 10]
[–10, 10] by [–10, 20]
[–15, 10] by [–30, 10]
[–7, 3] by [–50, 30]
[–10, 10] by [–10, 10][–5, 5] by [–5, 5]
ADDITIONAL ANSWERS 941
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 941
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84. (a) h � �16t2 � 170t � 6 (b) When t � 5.3125, 85. (a) V �
x(30 � 2x)(70 � 2x) in3
h � 457.5625. (b) Either x � 4.57 or x � 8.63 in.86. (a) &
(b)
86. (d) The two pilings are made of different materials, for
example.
87. (b) (c) The largest volume occurs when x � 70 (so it is
actually a sphere).
This volume is �43
��(70)3 � 1,436,755 ft3.
88. (a) y � 18.694x2 � 88.144x � 2393.0222 (b) y � �0.291x4 �
7.100x3 � 35.865x2 � 48.971x � 2336.634
(c) Using quadratic regression: $5768; Using quartic regression:
$3949
89. (a) y � 1.401x � 4.331 (b) y � 0.188x2 � 1.411x � 13.331 (c)
Using linear regression: In 2008;Using quadratic regression: In
2003
92. (a) R2 � �x1�
.21x.2
� (b) 2 ohms 93. (a) C(x) � �50
5�
0x
� (b) about 33.33 ounces of distilled water (c) x � �1030
� � 33.33
94. (a) S � 2�x2 � �20x00� (b) Either x � 2.31 cm and h � 59.75
cm, or x � 10.65 cm and h � 2.81 cm.
94. (c) Approximately 2.31 x 10.65 (graphically) and 2.81 h
59.75.
95. (a) S � x2 � �40x00� (b) 20 ft by 20 ft by 2.5 ft or x �
7.32, giving approximate dimensions 7.32 by 7.32 by 18.66.
94. (c) 7.32 x 20 (lower bound approximate), so y must be
between 2.5 and about 18.66.
[0, 15] by [0, 30][0, 15] by [0, 30]
[0, 15] by [0, 4500][0, 15] by [0, 4500]
[0, 70] by [0, 1,500,000]
[0, 255] by [0, 2.5]
[0, 11] by [0, 500]
942 ADDITIONAL ANSWERS
[0, 50] by [0, 1]
5144_Demana_TE_Ans_pp915-943 1/24/06 8:25 AM Page 942
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Chapter 2 ProjectAnswers are based on the sample data shown in
the table.
1. 2. y � �4.962(x � 1.097)2 � 0.8303. The sign of a affects the
direction the parabola opens.
The magnitude of a affects the vertical stretch of the
graph.Changes to h cause horizontal shifts to the graph, while
changes to k cause vertical shifts.
4. y � � 4.962x2 � 10.887x � 5.1415. y � � 4.968x2 � 10.913x �
5.1606. y � � 4.968 (x � 1.098)2 � 0.833
SECTION 3.1 Exploration 11. (0, 1) is in common; Domain: (��,
�); Range: (0, �); Continuous; Always increasing; Not symmetric; No
local extrema; Bounded below by
y � 0, which is also the only asymptote; limx → �
f(x) � �. limx → ��
f(x) � 0
2. (0, 1) is in common; Domain: (��, �); Range: (0, �);
Continuous; Always decreasing; Not symmetric; No local extrema;
Bounded below by y� 0, which is also the only asymptote; lim
x → �g(x) � 0, lim
x → ��g(x) � �
Exploration 21.
Exercises 3.118. Reflect f(x) � 2x over the y-axis and then
shift by 5 units to the right. 19. Vertically stretch f(x) � 0.5x
by a factor of 3 and then shift 4units up. 20. Vertically stretch
f(x) � 0.6x by a factor of 2 and then horizontally shrink by a
factor of 3. 21. Reflect f(x) � ex across they-axis and
horizontally shrink by a factor of 2. 22. Reflect f(x) � ex across
the x-axis and y-axis. Then, horizontally shrink by a factor of
3.23. Reflect f(x) � ex across the y-axis, horizontally shrink by a
factor of 3, translate 1 unit to the right and vertically stretch
by a factor of 2.24. Horizontally shrink f(x) � ex by a factor of
2, vertically stretch by a factor of 3, and shift down one unit.
25. Graph (a) is the only graphshaped and positioned like the graph
of y � bx, b � 1. 29. Graph (b) is the graph of y � 3�x translated
down 2 units. 30. Graph (f) is thegraph of y � 1.5x translated down
2 units. 31. Exponential decay; lim
x → �f(x) � 0, lim
x → ��f(x) � �
40. y2 � y3 since 2 � 23x � 2 � 21 � 23x � 2 � 21 � 3x � 2 � 23x
� 1
41. 42. 43. 44.
y-intercept: (0, 4) y-intercept: (0, 3) y-intercept: (0, 4)
y-intercept: (0, 3) Horizontal asymptotes: Horizontal asymptotes:
Horizontal asymptotes: Horizontal asymptotes:y � 0, y � 12 y � 0, y
� 18 y � 0, y � 16 y � 0, y � 9
[–5, 10] by [–5, 10][–5, 10] by [–5, 20][–5, 10] by [–5,
20][–10, 20] by [–5, 15]
[–4, 4] by [–2, 8]
[0, 1.6] by [–0.1, 1]
ADDITIONAL ANSWERS 943
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