5.1 Assess Your Understanding (page 346)
7. smooth; continuous 8. touches 9. 1-1, 12; 10, 02; 11, 12 10. r is a real zero of f; r is an x-intercept of the graph of f; x-r is a factor of f.
11. turning points 12. q; q 13. q; - q 14. As x increases in the positive direction, f1x2 decreases without bound. 15. b 16. d 17. Yes; degree 3; f1x2 =x3+4x; leading term: x3; constant term: 0 19. Yes; degree 2; g1x2 = -1
2 x2+1
2; leading term: -1
2 x2; constant term: 1 2 21. No; x is raised to the -1 power 23. No; x is raised to the 3
2 power 25. Yes; degree 4; F1x2 =5x4-px3+1
2; leading term: 5x4; constant term: 1 2 27. Yes; degree 4; G1x2 =2x4-4x3+4x2-4x+2; leading term: 2x4; constant term: 2
29. 31. 33. 35. 37.
39. 41.
y
x 5
5 (0, 1) (2, 1)
(1, 0)
y 5
x (1, 2)5 (1, 4)
(0, 3)
y 5
x (0, 0) 5
1,1 1,1 2
2
y 5
x 5 (1, 1)
(1, 1) (0, 0)
y 5
x 5 (2, 3) (1, 2) (0, 1)
y 5
x 5 (1, 1) (2, 3) (0, 3)
y 6
x 5 (3, 3) (2, 4) (1, 5)
43. f(x)=(x+1)(x -1)(x-3) =x3-3x2-x+3 for a=1
45. f(x)=x(x+3)(x -4) =x3-x2-12x for a=1 47. f(x)=(x+4)(x+1)(x -2)(x-3)
=x4-15x2+10x+24 for a=1
49. f(x)=(x+1)(x -3)2
=x3-5x2+3x+9 for a=1 51. f(x)=2(x+3)(x-1)(x -4)
=2x3-4x2 -22x+24
53. f(x)=16x(x+1)(x-2)(x -4) =16x4-80x3+32x2+128x
55. f(x)=5(x+1)2(x-1)2 =5x4-10x2+5
57. (a) 7, multiplicity 1; -3, multiplicity 2 (b) Graph touches the x-axis at -3 and crosses it at 7. (c) 2 (d) y=3x3 59. (a) 2, multiplicity 3 (b) Graph crosses the x-axis at 2. (c) 4 (d) y=4x5 61. (a) -1
2 , multiplicity 2; -4, multiplicity 3 (b) Graph touches the x-axis at -1
2 and crosses at -4. (c) 4 (d) y= -2x5 63. (a) 5, multiplicity 3; -4, multiplicity 2
(b) Graph touches the x-axis at -4 and crosses it at 5. (c) 4 (d) y=x5 65. (a) No real zeros (b) Graph neither crosses nor touches the x-axis.
(c) 5 (d) y =3x6 67. (a) 0, multiplicity 2; -22, 22 , multiplicity 1 (b) Graph touches the x-axis at 0 and crosses at -22 and 22 . (c) 3 (d) y= -2x4 69. Could be; zeros: -1, 1, 2; Least degree is 3. 71. Cannot be the graph of a polynomial; gap at x= -1 73. f1x2 =x1x-12 1x-22 75. f1x2 = -1
21x+12 1x-1221x-22 77. f1x2 =0.21x+42 1x+1221x-32 79. f1x2 = -x1x+3221x-322
81. Step 1: y=x3
Step 2: x-intercepts: 0, 3;
y-intercept: 0
Step 3: 0: multiplicity 2; touches;
3: multiplicity 1; crosses Step 4:
Step 5: (2, -4); (0, 0) Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on (- q, 0] and [2, q) Decreasing on [0, 2]
83. Step 1: y= -x3
Step 2: x-intercepts: -4, 1;
y-intercept: 16
Step 3: -4: multiplicity 2, touches;
1: multiplicity 1, crosses Step 4:
Step 5: 1-4, 02; 1-0.67, 18.522 Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on 3-4, -0.674 Decreasing on 1- q, -44 and 3-0.67, q2
85. Step 1: y = -2x4
Step 2: x-intercepts: -2, 2;
y-intercept: 32
Step 3: -2: multiplicity 1, crosses;
2: multiplicity 3, crosses Step 4:
Step 5: (-1, 54) Step 6:
Step 7: Domain: (- q, q);
Range: (- q, 54]
Step 8: Increasing on (- q, -1]
Decreasing on [-1, q)
10 2
10
4
y 24
x 5 (4, 16) (3, 0) (0, 0)
(2, 4) (1, 4)
60
7
60 5
y 50
50 x (0, 16)
(1, 0)
(2, 36) (4, 0)
(5, 6) (0.67, 18.52)
6 2
75 3
100
4
y
240 x 5 (2, 0) (1, 54) (0, 32)
(2, 0) (3, 10) (3, 250)
87. Step 1: y=x3
Step 2: x-intercepts: -4, -1, 2;
y-intercept: -8
Step 3: -4, -1, 2: multiplicity 1, crosses Step 4:
Step 5: (-2.73, 10.39); (0.73, -10.39) Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on (- q, -2.73]
and [0.73, q)
Decreasing on [-2.73, 0.73]
15
5
15
5
y 50
x 5 (3, 28)
(2, 0) (0.73, 10.39) (4, 0)
(1, 0) (5, 28)
(0, 8) (2.73, 10.39)
89. Step 1: y=x4
Step 2: x-intercepts: -2, 0, 2;
y-intercept: 0
Step 3: -2, 2: multiplicity 1, crosses;
0: multiplicity 2, touches Step 4:
Step 5: (-1.41, -4); (1.41, -4); (0, 0) Step 6:
Step 7: Domain: (- q, q);
Range: [-4, q)
Step 8: Increasing on [-1.41, 0] and [1.41, q) Decreasing on (- q, -1.41], and [0, 1.41]
91. Step 1: y=x4
Step 2: x-intercepts: -1, 2;
y-intercept: 4
Step 3: -1, 2: multiplicity 2, touches Step 4:
Step 5: (-1, 0); (2, 0); (0.5, 5.06) Step 6:
Step 7: Domain: (- q, q);
Range: [0, q)
Step 8: Increasing on [-1, 0.5] and [2, q) Decreasing on (- q, -1] and [0.5, 2]
10
3
4
3
y 90
x 5 (3, 45) (2, 0)(0, 0) (1.41, 4)
(2, 0) (1.41, 4)
(3, 45)
10
4
5
4
y 20
x (2, 0)5
(0.5, 5.06) (0, 4)
(1, 0)
(3, 16) (2, 16)
Step 5: (-2.37, -4.85); (-0.63, 0.35); (0, 0)
Step 6: y
x 5 5 5 (4, 48) 40
(0, 0) (1, 8) (3, 0)
(1, 0)
(0.63, 0.35)
(2.37, 4.85)
93. Step 1: y=x4
Step 2: x-intercepts: -1, 0, -3; y-intercept: 0 Step 3: -1, -3; multiplicity 1, crosses;
0: multiplicity 2, touches Step 4:
15
4
10 2
Step 7: Domain: (- q, q);
Range: [-4.85, q)
Step 8: Increasing on 3-2.37, -0.634 and [0, q)
Decreasing on (- q, -2.37] and [-0.63, 0]
Step 5:
(-2.57, -14.39); (-0.93, 30.18); (1.25, -64.75) Step 6:
Step 7: Domain: (- q, q); Range: [-64.75, q) Step 8: Increasing on 3-2.57, -0.934
and [1.25, q)
Decreasing on (- q, -2.574 and 3-0.93, 1.254
Step 5: (0, 0); (1.48, -5.91) Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on (- q, 0] and [1.48, q) Decreasing on [0, 1.48]
(2, 0) (0, 0) (0.93, 30.18)
(2, 0) (3, 0)
(1.25, 64.75) y
2x 3
60 40
(2.57, 14.39)
y 160
x 5 (0, 0) (2, 0)
(3, 108)
(1, 12)
(1.48, 5.91)
Step 6:
Step 7: Domain: 1- q, q2; Range: 1- q, q2 Step 8: Increasing on 1- q, -0.804 and 30.66, q2
Decreasing on 3-0.80, 0.664 99. Step 1: y=x3
Step 2:
Step 3: x-intercepts: -1.26, -0.20, 1.26;
y-intercept: -0.31752 Step 4:
Step 5: 1-0.80, 0.572; 10.66, -0.992
2
2
2
2
y 2.5
2.5x (1.5, 1.13) (1.26, 0) (0.66, 0.99) (0.5, 0.40)
(0.80, 0.57)
(0.20, 0) (1.26, 0)
(1.5, 0.86) (0, 0.32)
101. Step 1: y=x3 Step 2:
Step 3: x-intercepts: -3.56, 0.50;
y-intercept: 0.89 Step 4:
Step 5: 1-2.21, 9.912; 10.50, 02 Step 6:
Step 7: Domain: 1- q, q2; Range: 1- q, q2
Step 8: Increasing on 1- q, -2.214 and 30.50, q2
Decreasing on 3-2.21, 0.504
103. Step 1: y=x4 Step 2:
Step 3: x-intercepts: -1.5, -0.5, 0.5, 1.5;
y-intercept: 0.5625 Step 4:
Step 5: 1-1.12, -12; 11.12, -12, 10, 0.562 Step 6:
Step 7: Domain: 1- q, q2; Range: 3-1, q2
Step 8: Increasing on 3-1.12, 04 and 31.12, q2 Decreasing on 1- q, -1.124
and 30, 1.124
105. Step 1: y=2x4 Step 2:
Step 3: x-intercepts: -1.07, 1.62;
y-intercept: -4 Step 4:
Step 5: 1-0.42, -4.642 Step 6:
Step 7: Domain: 1- q, q2; Range: 3-4.64, q2 Step 8: Increasing on 3-0.42, q2
Decreasing on 1- q, -0.424
10 4
10 2
y 10
x 1 (1, 5.76)
(1, 1.14) (0.5, 0) (0, 0.89) (2.21, 9.91)
(3.56, 0) (3.9, 6.58)
3
2
2
2
y 5
x (1.5, 0) 2.5
(0.5, 0) (1.75, 2.29)
(1, 0.94)
(1.12, 1) (1.12, 1)
(1, 0.94) (1.75, 2.29) (1.5, 0)
(0.5, 0) (0, 0.56)
6
2
5
2
y 5
x 2 (1.25, 4.22)
(1.07, 0) (0.42, 4.64) (0, 4)
(1.75, 1.83) (1.62, 0)
95. Step 1: y=5x4
Step 2: x-intercepts: -3, -2, 0, 2;
y-intercept: 0
Step 3: -3, -2, 0, 2: multiplicity 1, crosses Step 4:
97. Step 1: y=x5
Step 2: x-intercepts: 0, 2; y-intercept: 0 Step 3: 0: multiplicity 2, touches;
2: multiplicity 1, crosses Step 4:
50 4
70 3
30
3
30
3
107. f (x)= -x(x + 2)(x -2) Step 1: y= -x3
Step 2: x-intercepts: -2, 0, 2;
y-intercept: 0
Step 3: -2, 0, 2: multiplicity 1, crosses Step 4:
Step 5: (-1.15, -3.08); (1.15, 3.08) Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on [-1.15, 1.15]
Decreasing on (- q, -1.15] and [1.15, q)
109. f (x) = x(x + 4) (x - 3) Step 1: y = x3
Step 2: x-intercepts: -4, 0, 3;
y-intercept: 0
Step 3: -4, 0, 3: multiplicity 1, crosses Step 4:
Step 5: (-2.36, 20.75); (1.69, -12.60) Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on (- q, -2.36]
and [1.69, q)
Decreasing on [-2.36, 1.69]
111. f (x) = 2x(x + 6)(x - 2)(x + 2) Step 1: y = 2x4
Step 2: x-intercepts: -6, -2, 0, 2;
y-intercept: 0
Step 3: -6, -2, 0, 2: multiplicity 1, crosses Step 4:
Step 5: (-4.65, -221.25); (-1.06, 30.12);
(1.21, -44.25) Step 6:
Step 7: Domain: (- q, q);
Range: [-221.25, q)
Step 8: Decreasing on (- q, -4.65] and [-1.06, 1.21]
Increasing on [-4.65, -1.06] and [1.21, q)
20
5
20
5
y 20
x 5 (3, 15) (2, 0) (0, 0)
(1.15, 3.08) (2, 0)
(3, 15)
(1.15, 3.08)
32 5
40
5
y 50
x 5 (4, 0)
(2.36, 20.75)
(5, 40) (0, 0) (1.69, 12.60) (4, 32) (3, 0)
150 7
225 3
y 700
x 3 (2, 0) (6, 0)
(1.06, 30.12) (7, 630)
(4.65, 221.25) (1.21, 44.25)
(2, 0) (3, 270)
(0, 0)
113. f (x) =-x2(x+1)2(x-1) Step 1: y= -x5
Step 2: x-intercepts: -1, 0, 1;
y-intercept: 0
Step 3: 1: multiplicity 1, crosses; -1, 0: multiplicity 2, touches
Step 4:
Step 5: (-1, 0); (-0.54, 0.10); (0, 0); (0.74, 0.43)
Step 6:
Step 7: Domain: (- q, q);
Range: (- q, q)
Step 8: Increasing on [-1, -0.54] and [0, 0.74]
Decreasing on (- q, -1], [-0.54, 0], and [0.74, q)
1
2
1
2
y 2.5
x (0.54, 0.10) 2.5
(1, 0) (1.5, 1.40625)
(1.2, 1.39392) (1, 0) (0, 0)
(0.74, 0.43)
115. f1x2 =31x+32 1x-12 1x-42 117. f1x2 = -21x+5221x-22 1x-42 119. (a) -3, 2 (b) -6, -1 121. (a)
The relation appears to be cubic.
(b) H1x2 =0.3948x3-5.9563x2+26.1965x -7.4127 (c) ≈24 (d)
(e) ≈54; no. The end behavior of the model indicates that as time goes on, the number of major hurricanes will continue to increase each decade without limit. This is unrealistic. End behavior should not be used to make predictions too far outside the data used to create the model.
123. (a)
The relation appears to be cubic.
(b) 6°/h (c) 0.17°/h
(d) T(x)= -0.01992x2+0.6745x2-4.4360x+48.4643; 70.1°F (e)
(f) The predicted temperature at midnight is 48.5°F.
H
x 50
40 30 20 10
0 2 4 6 8 10
35
00 10
T
x 70
62 54 46 38 30
0 6 12 18 24
74
030 27
125. (a) (b)
(c) (d) As more terms are added, the values of the polynomial function get closer to the values of f. The approximations near 0 are better than those near -1 or 1.
131. 1a2-1d2 135. y= -2 5 x- 11
5 136. 5xx≠ -56 137. -2-27
2 , -2+ 27
2 138. e-4 5, 2f Historical Problems (page 363)
1. ax- b
3b3+bax- b
3b2+cax- b
3b +d=0 x3-bx2+b2x
3 - b3
27+bx2 - 2b2x 3 + b3
9 +cx - bc 3 +d=0 x3+ ac- b2
3bx+ a2b3 27 - bc
3 +db =0 Let p=c - b2
3 and q=2b3 27 - bc
3 +d. Then x3+px+q=0.
2. (H+K)3+p(H+K)+q=0
H3+3H2K+3HK2+K3+pH+pK+q=0 Let 3HK= -p.
H3-pH-pK+K3+pH+pK+q=0, H3+K3= -q
3. 3HK= -p
K= - p 3H H3+ a- p
3Hb3= -q H3- p3
27H3= -q 27H6- p3= -27qH3 27H6+27qH3- p3=0
H3= -27q { 2(27q)2-4(27)(-p3) 2#27
H3= -q
2 {
B 272q2
22(272)+ 4(27)p3 22(272) H3= -q
2 {
B q2
4 + p3 27 H=
B3 -q
2 + B q2
4 + p3 27 Choose the positive root for now.
4. H3+K3= -q K3= -q-H3 K3= -q- c-q
2 + B
q2 4 + p3
27d K3= -q
2 - B
q2 4 + p3
27 K=
B
3 -q 2 - B
q2 4 + p3
27
5. x=H+K x=
B
3 -q 2 +
B q2
4 + p3 27 +
B
3 -q 2 -
B q2
4 + p3 27
(Note that had we used the negative root in 3 the result would be the same.) 6. x=3 7. x=2 8. x=2
5.2 Assess Your Understanding (page 363)
5. a 6. f1c2 7. b 8. F 9. 0 10. T 11. R=f (2)=8; no 13. R=f (2)=0; yes 15. R=f (-3)=0; yes 17. R=f (-4)=1; no 19. R=f a1
2b =0; yes 21. 7; 3 or 1 positive; 2 or 0 negative 23. 6; 2 or 0 positive; 2 or 0 negative 25. 3; 2 or 0 positive; 1 negative 27. 4; 2 or 0 positive; 2 or 0 negative 29. 5; 0 positive; 3 or 1 negative 31. 6; 1 positive; 1 negative 33. 4; {1, {1
3 35. 5; {1, {3 37. 3; {1, {2, {1
4, {1
2 39. 4; {1, {3, {9, {1 2, {1
3, {1 6, {3
2, {9
2 41. 5; {1, {2, {3, {4, {6, {12, {1 2, {3
2 43. 4; {1, {2, {4, {5, {10, {20, {1
2, {5 2, {1
3, {2 3, {4
3, {5 3, {10
3, {20 3, {1
6, {5 6
2
1
2
1
200
11
200
11
100
9
50 3
45. -1 and 1 47. -11 and 11 49. -9 and 3
51. -3, -1, 2; f (x)=(x+3)(x+1)(x -2) 53. 1
2, 3, 3; f(x)=(2x -1)(x-3)2 55. - 1
3; f (x)=(3x+1)(x2+x+1) 57. 4, 3- 25, 3+ 25; f(x)=(x-4)(x -3+ 25)(x-3- 25) 59. -2, -1, 1, 1; f (x)=(x+2)(x +1)(x-1)2 61. -7
3, -1, 2
7, 2; f (x) =(x-2)(x+1)(3x+7)(7x-2) 63. 3, 5+ 217
2 , 5- 217
2 ; f (x)=(x-3)ax- 5+ 217
2 b ax-5- 217
2 b
65. -1 2, 1
2; f (x)=(2x+1)(2x-1)(x2+2) 67. 22 2 , -22
2 , 2; f (x)=(x-2)(22x-1)(22x+1)(2x2+1) 69. -5.9, -0.3, 3 71. -3.8, 4.5 73. -43.5, 1, 23 75. 5-1, 2 6 77. e2
3, -1+ 22, -1- 22f 79. e1
3, 25, -25f 81. 5-3, -26 83. e-1 3f 85. f (0)= -1; f (1)=10; Zero: 0.21 87. f (-5)= -58; f (-4)=2; Zero: -4.04 89. f (1.4)= -0.17536; f (1.5)=1.40625; Zero: 1.41 91. Step 1: y=x3
Step 2: x-intercepts: -3, -1, 2;
y-intercept: -6
Step 3: -3, -1, 2: multiplicity 1, crosses
Step 4:
Step 5: (-2.12, 4.06); (0.79, -8.21)
Step 6:
Step 7: Domain and range: (- q, q) Step 8: Increasing: (- q, -2.12], [0.79, q)
Decreasing: [-2.12, 0.79]
20
4
15 3
y 30
5 x (1, 0)
(2.12, 4.06) (3, 0)
(4, 18) (0.79, 8.21)
(0, 6) (2, 0)
(3, 24)
93. Step 1: y=x4
Step 2: x-intercepts: -2, -1, 1;
y-intercept: 2
Step 3: -2, -1: multiplicity 1, crosses;
1: multiplicity 2, touches
Step 4:
Step 5: (-1.59, -1.62), (-0.16, 2.08), (1, 0)
Step 6:
Step 7: Domain: 1- q, q);
Range: [-1.62, q)
Step 8: Increasing: [-1.59, -0.16], [1, q) Decreasing: (- q, -1.59], [-0.16, 1]
10
3
2 2
y 16
x 5 (1, 0)
(2, 0)
(1.59, 1.62) (0.16, 2.08) (1, 0) (0, 2) (2, 12)
95. Step 1: y=4 x5
Step 2: x-intercepts: -22 2 , 22
2 , 2;
y-intercept: 2 Step 3: -22
2 , 22
2 , 2: multiplicity 1, crosses
Step 4:
Step 5: (-0.30, 2.23); (1.61, -10.09)
Step 6:
Step 7: Domain and range: (- q,q) Step 8: Increasing: 1- q, -0.30], [1.61, q)
Decreasing: [-0.30, 1.61]
10 2
12
3
y 4
x 5
(1, 9) (1, 3)
(1.61, 10.09) (0, 2) (2, 0) (0.30, 2.23) , 0
2 2
2 , 0 2
97. Step 1: y=6x4 Step 2: x-intercepts: -1
2, 2 3, 3;
y-intercept: -18 Step 3: -1
2, 2
3 : multiplicity 1, crosses;
3: multiplicity 2, touches
Step 4:
Step 5: (-0.03, -18.04), (1.65, 23.12), (3, 0)
Step 6:
Step 7: Domain: (- q,q);
Range: [-18.04,q)
Step 8: Increasing: [-0.03, 1.65], [3, q);
Decreasing: (- q, -0.03], [1.65, 3]
25
1 20
4
y 25
x 5 (0.03, 18.04)
(0, 18) (1.65, 23.12)
(3, 0) 2, 3 0 1,
2 0
99. k=5 101. -7 103. If f (x)=xn-cn, then f (c)=cn-cn=0; so x-c is a factor of f. 105. 5 107. 7 in.
109. All the potential rational zeros are integers, so r is either an integer or is not a rational zero (and is, therefore, irrational). 111. 0.215 113. No; by the Rational Zeros Theorem, 1
3 is not a potential rational zero. 115. No; by the Rational Zeros Theorem, 2
3 is not a potential rational zero.
116. y=2 5x-3
5 117. 33, 82 118. (0, -223), (0, 223), (4, 0) 119. 3-3, 24 and 35, q) 5.3 Assess Your Understanding (page 370)
3. one 4. 3-4i 5. T 6. F 7. 4+i 9. -i, 1-i 11. -i, -2i 13. -i 15. 2-i, -3+i 17. f (x)=x4-14x3+77x2-200x+208; a=1 19. f (x)=x5-4x4+7x3-8x2+6x-4; a=1 21. f (x)=x4-6x3+10x2-6x+9; a=1 23. -2i, 4 25. 2i, -3, 1
2 27. 3+2i, -2, 5 29. 4i, -211, 211, -2
3 31. 1, -1 2 - 23
2 i, -1 2+ 23
2 i; f (x)=(x-1)ax+ 1 2+ 23
2 ib ax+ 1 2- 23
2 ib 33. 2, 3-2i, 3+2i; f (x)=(x-2)(x-3+2i)(x-3-2i) 35. -i, i, -2i, 2i; f (x)=(x+i)(x -i)(x+2i)(x-2i)
37. -5i, 5i, -3, 1; f (x)=(x+5i)(x-5i)(x+3)(x -1) 39. -4, 1
3, 2-3i, 2+3i; f (x)=3(x+4)ax-1
3b(x-2+3i)(x-2-3i) 41. 130 43. (a) f1x2 = 1x2- 22x+12 1x2+ 22x+12 (b) -22
2 -22 2 i, -22
2 + 22 2 i, 22
2 -22 2 i, 22
2 + 22 2 i
45. Zeros that are complex numbers must occur in conjugate pairs; or a polynomial with real coefficients of odd degree must have at least one real zero.
47. If the remaining zero were a complex number, its conjugate would also be a zero, creating a polynomial of degree 5.
49. 50. 5-226 51. 6x3-13x2-13x+20 52. A=9p ft2 1 about 28.274 ft22; C=6p ft 1 about 18.850 ft2
5.4 Assess Your Understanding (page 379)
5. F 6. horizontal asymptote 7. vertical asymptote 8. proper 9. T 10. F 11. y=0 12. T 13. d 14. a 15. All real numbers except 3; 5x0x≠36 17. All real numbers except 2 and -4; 5x0x≠2, x≠ -46
19. All real numbers except -1
2 and 3; ex`x≠ -1
2, x≠3f 21. All real numbers except 2; 5x0x≠26 23. All real numbers 25. All real numbers except -3 and 3; 5x0x≠ -3, x≠36
27. (a) Domain: 5x0x≠26; range: 5y0y≠16 (b) 10, 02 (c) y=1 (d) x=2 (e) None
29. (a) Domain: 5x0x≠06; range: all real numbers (b) 1-1, 02, 11, 02 (c) None (d) x=0 (e) y=2x 31. (a) Domain: 5x0x≠ -2, x≠26; range: 5y0y…0, y716 (b) 10, 02 (c) y=1 (d) x= -2, x=2 (e) None
y 7
5
x 10 2
33. (a)
(b) Domain: 5xx≠06; range: 5yy≠26 (c) Vertical asymptote: x=0;
horizontal asymptote: y=2
35. (a)
(b) Domain: 5xx≠16; range: 5yy706 (c) Vertical asymptote: x=1;
horizontal asymptote: y=0
37. (a)
(b) Domain: 5xx≠ -16; range: 5yy≠06 (c) Vertical asymptote: x= -1;
horizontal asymptote: y=0 39. (a)
(b) Domain: 5xx≠ -26; range: 5yy606 (c) Vertical asymptote: x= -2;
horizontal asymptote: y=0
41. (a)
(b) Domain: 5xx≠36; range: 5yy716 (c) Vertical asymptote: x=3;
horizontal asymptote: y=1
43. (a)
(b) Domain: 5xx≠06; range: 5yy616 (c) Vertical asymptote: x=0;
horizontal asymptote: y=1
y 8
5 x (1, 1)
(1, 3) y 2
x 0
y 10
x 5 (0, 1) (2, 1) y 0 x 1
y 5
5 x (2, 2)
(1, 1) x 1 y 0
y 2
x 4 (1, 1) (3, 1)
x 2 y 0
y 10
x 9 (4, 3) (2, 3)
x 3
y 1
y 3
x (2, 0) 5
(2, 0)
x 0 y 1
45. Vertical asymptote: x= -4; horizontal asymptote: y=3 47. Vertical asymptote: x=3; oblique asymptote: y=x+5 49. Vertical asymptotes: x=1, x= -1; horizontal asymptote: y=0 51. Vertical asymptote: x= -1
3; horizontal asymptote: y=2 3 53. No asymptotes 55. Vertical asymptote: x=0; no horizontal or oblique asymptote
57. (a) 9.8208 m/sec2 (b) 9.8195 m/sec2 (c) 9.7936 m/sec2 (d) y=0 (e) ∅ 59. (a)
(b) Horizontal: Rtot =10; as the resistance of R2 increases without bound, the total resistance approaches 10 ohms, the resistance R1 .
(c) R1 ≈103.5 ohms
61. (a) R1x2 =2+ 5
x-1 =5a 1 x-1b +2 (b)
(c) Vertical asymptote: x=1;
horizontal asymptote: y=2
67. x=5 68. e- 4 19f 69. x@axis symmetry 70. 1-3, 112, 12, -42
Rtot
R2 5
10
0 5 10 15 20 25
y
x 10
10 5 5
x 1
y 2 y 2 (0, 3)
(2, 7)
5.5 Assess Your Understanding (page 390) 2. False 3. c 4. a 5. (a) 5x0x≠26 (b) 0 6. True 7. Step 1: Domain: 5x0x≠0, x≠ -46
Step 2: R is in lowest terms
Step 3: no y-intercept; x-intercept: -1 Step 4: R is in lowest terms;
vertical asymptotes: x=0, x= -4 Step 5: Horizontal asymptote: y=0,
intersected at (-1, 0) Step 6:
Step 7:
9. Step 1: R(x)= 3(x+1) 2(x+2); domain: 5x0x≠ -26 Step 2: R is in lowest terms Step 3: y-intercept: 3
4; x-intercept: -1 Step 4: R is in lowest terms;
vertical asymptote: x =-2 Step 5: Horizontal asymptote: y= 3
2, not intersected
Step 6:
Step 7:
11. Step 1: R(x)= 3 (x+2)(x-2); domain: 5x0x≠ -2, x≠26 Step 2: R is in lowest terms Step 3: y-intercept: -3
4; no x-intercept Step 4: R is in lowest terms;
vertical asymptotes: x=2, x= -2 Step 5: Horizontal asymptote: y=0, not
intersected Step 6:
Step 7:
4
7
4 3
y 2.5
4 x
(1, 0) x 4x 0
y 0 4
5, 5 2 , 7 1 2 2,1
4
1, 2 5
8
6
4 4
y 5
x 5 (1, 0) (3, 3)
x 2
y
3
, 2 3 2
0,3 4 3 2
5 5
5
5
y 2
x 5 x 2 x 2
y 0 3,3
5 3,3
5 3
0, 4
15. Step 1: H(x)=(x-1)(x2 +x+1) (x+3)(x -3) ; domain: 5x0x≠ -3, x≠36 Step 2: H is in lowest terms Step 3: y-intercept: 1
9; x-intercept: 1 Step 4: H is in lowest terms; vertical
asymptotes: x=3, x= -3 Step 5: Oblique asymptote: y=x,
intersected at a1 9, 1
9b Step 6:
Step 7:
13. Step 1: P(x)=(x2+x+1)(x2-x+1) (x+1)(x-1) ; domain: 5x0x≠ -1, x≠16 Step 2: P is in lowest terms
Step 3: y-intercept: -1; no x-intercept Step 4: P is in lowest terms;
vertical asymptotes: x= -1, x=1 Step 5: No horizontal or oblique asymptote Step 6:
Step 7:
17. Step 1: R(x)= x2 (x+3)(x-2); domain: 5x≠ -3, x≠26 Step 2: R is in lowest terms Step 3: y-intercept: 0; x-intercept: 0 Step 4: R is in lowest terms; vertical
asymptotes: x=2, x= -3 Step 5: Horizontal asymptote: y=1,
intersected at (6, 1) Step 6:
Step 7:
12
4
6
4
y 6
x 5 (0, 1)
(2, 7) (2, 7)
x 1 x 1
12 8
12
8
y 8
x (4, 9.3) 10
(2, 1.4) (4, 9)
(1, 0)
x 3 x 3 0,1 y x
9
6 8
6
8
y 2
x 10 (6, 1.5)
(1, 0.25)(0, 0) (3, 1.5)
x 3 x 2 y 1
1 1, 6
Step 6:
19. Step 1: G(x)= x (x+2)(x-2); domain: 5x0x≠ -2, x≠26 Step 2: G is in lowest terms Step 3: y-intercept: 0; x-intercept: 0 Step 4: G is in lowest terms;
vertical asymptotes: x= -2, x=2 Step 5: Horizontal asymptote: y=0,
intersected at (0, 0)
Step 7:
4 4
4
4
y 2
x 5 (0, 0)
x 2 x 2
y 0 1,1
3
3 3, 5
3,3 5
1 1, 3
21. Step 1: R(x)= 3
(x-1)(x+2)(x-2); domain: 5x0x≠1, x≠ -2, x≠26 Step 2: R is in lowest terms
Step 3: y-intercept: 3
4; no x-intercept Step 4: R is in lowest terms; vertical
asymptotes: x= -2, x=1, x=2 Step 5: Horizontal asymptote: y=0,
not intersected
Step 6: Step 7:
5
4
5
4
y 5
x 5
x 2 x 2
x 1
y 0 0,3
4 3 3, 20
3,3 10
24 , 7 3 2
23. Step 1: H(x)= (x+1)(x -1) (x2+4)(x +2)(x-2); domain: 5x0x≠ -2, x≠26 Step 2: H is in lowest terms Step 3: y-intercept: 1
16; x-intercepts: -1, 1 Step 4: H is in lowest terms;
vertical asymptotes: x= -2, x=2 Step 5: Horizontal asymptote: y=0,
intersected at (-1, 0) and (1, 0) Step 6:
Step 7:
25. Step 1: F(x)=(x+1)(x -4) x+2 ; domain: 5x0x≠ -26 Step 2: F is in lowest terms
Step 3: y-intercept: -2; x-intercepts: -1, 4 Step 4: F is in lowest terms;
vertical asymptote: x= -2 Step 5: Oblique asymptote: y=x-5,
not intersected Step 6:
Step 7:
27. Step 1: R(x) =(x+4)(x-3) x-4 ; domain: 5x0x≠46 Step 2: R is in lowest terms
Step 3: y-intercept: 3; x-intercepts: -4, 3 Step 4: R is in lowest terms;
vertical asymptote: x=4 Step 5: Oblique asymptote: y=x+5,
not intersected Step 6:
Step 7:
29. Step 1: F(x) =(x+4)(x -3) x+2 ; domain: 5x0x≠ -26 Step 2: F is in lowest terms
Step 3: y-intercept: -6; x-intercepts: -4, 3 Step 4: F is in lowest terms;
vertical asymptote: x= -2 Step 5: Oblique asymptote: y=x-1,
not intersected Step 6:
Step 7:
31. Step 1: Domain: 5x0x≠ -36 Step 2: R is in lowest terms
Step 3: y-intercept: 0; x-intercepts: 0, 1 Step 4: Vertical asymptote: x= -3 Step 5: Horizontal asymptote: y=1,
not intersected Step 6:
Step 7:
33. Step 1: R(x)=(x+4)(x-3) (x+2)(x-3); domain: 5x0x≠ -2, x≠36 Step 2: In lowest terms, R(x)=x+4
x+2 Step 3: y-intercept: 2; x-intercept: -4 Step 4: Vertical asymptote: x= -2;
hole at a3, 7 5b
Step 5: Horizontal asymptote: y=1, not intersected
Step 6:
Step 7:
1
4
1
4
y
x 5 0.3
(3, 0.12) (1, 0) (1.5, 0.11) (1.5, 0.11)
(1, 0) (3, 0.12)
0, 1 16
x 2 x 2 y 0
6 7
20
7
y 12
x 10 (4, 0)
(5, 0.86) (0, 2) (3, 14) (1.5, 5.5)
(1, 0)
x 2
y x 5
20
5 4
10
y 21
x 10 (5, 18) (0, 3)
(3, 0) (3.5, 7.5) (4, 0)
5, 8
9 x 4
y x 5
12 6
12
6
y 10
x (3, 0)10 (0, 6) (4, 0) (3, 6)
5, 8 3
x 2
y x 1 4, 4
3
15
15
15
15
0.01
0.5
0.01
2.5
y 10
10x
y 0.01
(1, 0) 1.25
(1, 0) (0, 0) (0, 0)
See enlarged view at right.
Enlarged view x
x 3
y 1
6
8
6 4
y
10x (0, 2)
(3, 1) (4, 0)
3, 7 5
4, 4 3 5, 1
3
x 2 y 1 5
35. Step 1: R(x)=(3x+1)(2x-3) (x-2)(2x-3); domain: ex`x≠3
2, x≠2f Step 2: In lowest terms, R(x)=3x+1
x-2 Step 3: y-intercept: -1
2; x-intercept: -1 3 Step 4: Vertical asymptote: x=2;
hole at a3 2, -11b
Step 5: Horizontal asymptote: y=3, not intersected
Step 6: Step 7:
10
3 5
10
y 8
x 10 (6, 4.75) x 2 y 3
1,2 3
, 0 1
3 1 0, 2
3, 11 2
37. Step 1: R(x)=(x+3)(x+2) x+3 ; domain: 5x0x≠ -36 Step 2: In lowest terms, R(x)=x+2 Step 3: y-intercept: 2; x-intercept:-2 Step 4: Vertical asymptote: none;
hole at (-3, -1) Step 5: No horizontal or oblique
asymptote
Step 6: Step 7:
39. 1. H1x2 = -31x-22
1x-22 1x+22; domain: 5xx≠ -2, x≠26 2. In lowest terms, H1x2 = -3
x+2 3. y-intercept: -3
2; no x-intercept 4. Vertical asymptote: x = -2; hole at a2, -3
4b 5. Horizontal asymptote: y=0; not intersected 6. 7.
4 8
4 4
y 5
x 5 (2, 0) (3, 1)
(4, 2) (0, 2)
1
, 2 5 2
10
8
10 4
y 10
x
−7 (−3, 3) y = 0
x = −2
Q3, − R35 Q0, − R32 Q2, − 34R
41. 1. F1x2 = 1x-12 1x-42
1x-122 ; domain: 5xx≠16 2. In lowest terms, F1x2 =x-4
x-1 3. y-intercept: 4; x-intercept: 4 4. Vertical asymptote: x =1 5. Horizontal asymptote: y =1; not intersected
6. 7.
6
6
6
6
y 5
−5 x
−5 5
(0, 4)
(2, −2) y = 1
x = 1 Q5, R14 (4, 0)
43. 1. G1x2 = x
1x+222; domain: 5xx≠ -26 2. G is in lowest terms 3. y-intercept: 0; x-intercept: 0 4. Vertical asymptote: x= -2 5. Horizontal asymptote: y=0; intersected at (0, 0)
6. 7.
1 6
4
6
y 3
−7 x
−7 3
(−1, −1) (0, 0) (−3, −3)
x = −2 y = 0
Q1, R19
45. Step 1: f (x)=x2+1
x ; domain: 5x0x≠06 Step 2: f is in lowest terms
Step 3: no y-intercept; no x-intercepts Step 4: f is in lowest terms;
vertical asymptote: x=0 Step 5: Oblique asymptote: y=x,
not intersected Step 6:
Step 7:
47. Step 1: f (x) =x3+1
x = (x+1)(x2-x+1)
x ;
domain: 5x0x≠06 Step 2: f is in lowest terms
Step 3: no y-intercept; x-intercept:-1 Step 4: f is in lowest terms;
vertical asymptote: x=0
Step 5: No horizontal or oblique asymptote Step 6:
Step 7:
49. Step 1: f (x)=x4+1
x3 ; domain: 5x0x≠06 Step 2: f is in lowest terms
Step 3: no y-intercept; no x-intercepts Step 4: f is in lowest terms;
vertical asymptote: x=0 Step 5: Oblique asymptote: y=x,
not intersected Step 6:
Step 7:
5 5
5
5
y 5
x (1, 2)5 (1, 2)
x 0
y x
10 5
10
5
y 5
x 5 (1, 0)
(1, 2)
x 0 2,7
2
7
, 4 1 2
5
5
5
5
y 5
x 5 (1, 2)
(1, 2)
x 0 y x
51. One possibility: R1x2 = x2
x2-4 53. One possibility: R1x2 =
1x-12 1x-32 ax2+ 4 3b 1x+1221x-222 55.
The likelihood of your ball being chosen decreases very quickly and approaches 0 as the number of attendees, x, increases.
57. (a) t-axis; C1t2S0 (b)
(c) 0.71 h after injection
59. (a) C1x2 =16x+5000
x +100 (b) x70 (c)
(d) Approximately 17.7 ft by 56.6 ft (longer side parallel to river)
P(x)
x 1
0 10 203040 50 60
0.4
00 12
10,000
00 300
61. (a) S1x2 =2x2+40,000 x (b)
(c) 2784.95 in.2
(d) 21.54 in.*21.54 in.*21.54 in.
(e) To minimize the cost of materials needed for construction
63. (a) C1r2 =12pr2+4000 r (b)
The cost is smallest when r=3.76 cm.
10,000
00 60
6000
00 10
65. No. Each function is a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for x=1; each graph has a hole at x=1. 71. If there is a common factor between the numerator and the denominator, and the factor yields a real zero, then the graph will have a hole.
72. 4x3-5x2+2x+2 73. e- 1
10f 74. 17
2 75. ≈ -0.164 5.6 Assess Your Understanding (page 398)
3. c 4. F 5. (a) 5x006x61 or x726; 10, 12∪12, q2 (b) 5x0x…0 or 1…x…26; 1- q, 04∪31, 24 7. (a) 5x0-16x60 or x716; 1-1, 02∪11, q2 (b) 5x0x6 -1 or 0…x616; 1- q, -12∪30, 12
9. 5x0x60 or 0 6x636; 1- q, 02∪10, 32 11. 5xx…16; 1- q, 14 13. 5x0x… -2 or xÚ26; 1- q, -24∪32, q2
15. 5x0 -46x6 -1 or x706; 1-4, -12∪10, q2 17. 5x0-26x… -16; 1-2, -1] 19. 5x0x6 -26; 1- q, -22 21. 5x0x746; 14, q2 23. 5x0-46x60 or x706; 1-4, 02∪10, q2 25. 5x0x…1 or 2…x…36; 1- q, 1]∪32, 34 27. 5x0-16x60 or x 736; 1-1, 02∪13, q2 29. 5x0x6 -1 or x716; 1- q, -12∪11, q2 31. 5x0x6 -1 or x716; 1- q, -12∪11, q2 33. 5x0x6 -1 or x 716; 1- q, -12∪11, q2 35. 5x0x… -1 or 0 6x…16; 1- q, -1]∪10, 14 37. 5x0x6 -1 or x716; 1- q, -12∪11, q2 39. 5x0x626; 1- q, 22
41. 5x0-26x…96; 1-2, 94 43. 5x0x62 or 3 6x656; 1- q, 22∪13, 52
45. 5x0x6 -5 or -4…x… -3 or x=0 or x 716; 1- q, -52∪3-4, -34∪506∪11, q2 47. ex`-1
2 6x61 or x73f; a-1
2, 1b∪13, q2 49. 5x0-16x63 or x756; 1-1, 32∪15, q2 51. ex`x… -4 or xÚ 1
2 f; 1- q, -44∪c1
2, qb 53. 5x0x63 or xÚ76; 1- q, 32∪[7, q2 55. 5x0x626; 1- q, 22 57. ex`x6 -2
3 or 06x63
2 f; a- q, -2
3 b∪a0, 3
2 b 59. 5x0x… -3 or 0…x…36; 1- q, -34∪30, 34 61. (a) -4, -1
2, 3
(b) f(x)=(x+4)2(2x+1)(x-3) (c)
(d) a-1 2, 3b
y 75
x 5 (0, 48) (4, 0)
(1.74, 185.98) (3, 0) (1.87, 60.54)
, 0 2
1
63. (a)
(b) 1- q, -64∪31, 22∪12, q2
65. (a)
(b) 3-4, -22∪3-1, 32∪13, q2 67. 5x0x746; 14, q2
69. 5x0x… -2 or x Ú26; 1- q, -24∪32, q2 71. 5x0x6 -4 or x Ú26; 1- q, -42∪32, q2
y 10
x (1, 0)10 (6, 0)
, 3 2 0
,9 10 4 ,15 4 2
x 2 y 1
y 10
x 10 (4, 0)
(1, 0) (0, 2)
,28 3 5 y x 3
x 2
73.
f1x2 …g1x2 if -1…x…1
75.
f1x2 …g1x2 if -2…x…2
77. Produce at least 250 bicycles 79. (a) The stretch is less than 39 ft.
(b) The ledge should be at least 84 ft above the ground for a 150-lb jumper.
81. At least 50 students must attend. 86. c4 3, qb 87. x2-x-4 88. 3x2y41x+2y2 12x-3y2 89. y=2
32x
y 2.5
x (1, 0) (1, 0)2.5
f(x) x4 1
g(x) 2x2 2
y 32
x 2.5
(2, 12) (2, 12)
f(x) x4 4 g(x) 3x2
Review Exercises (page 402)
1. Polynomial of degree 5 2. Rational 3. Neither 4. Polynomial of degree 0
5. 6. 7.
8. Step 1: y=x3
Step 2: x-intercepts: -4, -2, 0; y-intercept: 0 Step 3: -4, -2, 0: multiplicity 1; crosses Step 4:
Step 5: (-3.15, 3.08), (-0.85, -3.08) Step 6:
Step 7: Domain: (- q, q); Range: (- q, q) Step 8: Increasing on (- q, -3.15] and
[-0.85, q)
Decreasing on [-3.15, -0.85]
y 15
x 2 (0, 8) (2, 0) (4, 8)
y 4
x 7 (1, 0)
(2, 1) (0, 1)
y 18
3 x (1, 2) (0, 3) (2, 3)
9. Step 1: y=x3
Step 2: x-intercepts: -4, 2; y-intercept: 16 Step 3: -4: multiplicity 1; crosses;
2: multiplicity 2; touches Step 4:
Step 5: (-2, 32), (2, 0) Step 6:
Step 7: Domain: (- q, q); Range: (- q, q) Step 8: Increasing on (- q, -2] and [2, q)
Decreasing on [-2, 2]
10. Step 1: y= -2x3
Step 2: x-intercepts: 0, 2; y-intercept: 0 Step 3: 0: multiplicity 2; touches;
2: multiplicity 1; crosses Step 4:
Step 5: (0, 0), (1.33, 2.37) Step 6 :
Step 7: Domain: (- q, q); Range: (- q, q) Step 8: Increasing on [0, 1.33]
Decreasing on (- q, 0] and [1.33, q)
15 6
15 2
y 20
2 (1, 15) (0, 0) (0.85, 3.08) (4, 0)
(3.15, 3.08) (2, 0)
(5, 15)
x
40 6
60 5
y
60 x 10 (3, 7) (0, 16)
(2, 0) (5, 49)
(4, 0) (2, 32)
10 5
10
5
y 20
6 (1.33, 2.37) (0, 0) (2, 0)
(3, 18) (1, 6)
x
11. Step 1: y=x4
Step 2: x-intercepts: -3, -1, 1;
y-intercept: 3
Step 3: -3,-1: multiplicity 1; crosses;
1: multiplicity 2; touches
Step 4:
Step 5: (-2.28,-9.91), (-0.22, 3.23), (1, 0)
Step 6:
Step 7: Domain: (- q, q); Range: [-9.91, q) Step 8: Increasing on [-2.28, -0.22] and [1, q)
Decreasing on (- q, -2.28] and [-0.22, 1]
80
4
20
4
y 80
x (1, 0)5 (0, 3)
(2, 15) (2.28, 9.91)
(3, 0) (1, 0)
(0.22, 3.23) (4, 75)
12. R=10; g is not a factor of f. 13. R=0; g is a factor of f. 14. f (4)=47,105 15. 4, 2, or 0 positive; 2 or 0 negative 16. 1 positive; 2 or 0 negative 17. {1, {3, {1
2, {3 2, {1
3, {1 4, {3
4, {1 6, {1
12 18. -2, 1, 4; f (x)=(x+2)(x -1)(x-4) 19. 1
2, multiplicity 2; -2; f (x)=4ax-1
2b2(x+2) 20. 2, multiplicity 2; f (x)=(x-2)2(x2+5) 21. 5-3, 26 22. e-3, -1, -1
2, 1f 23. -2 and 3 24. -5 and 5
20
3 10
4
40
5
10
5
25. f (0)= -1; f (1)=1 26. f (0)= -1; f (1)=1 27. 1.52 28. 0.93
29. 4-i; f (x)=x3-14x2+65x-102 30. -i, 1-i; f (x)=x4-2x3+3x2-2x+2 31. -2, 1, 4; f (x)=(x+2)(x -1)(x -4)
32. -2, 1
2 (multiplicity 2); f (x)=4(x+2)ax-1 2b2
33. 2 (multiplicity 2), -25i, 25i; f(x)=(x+ 25i)(x- 25i)(x-2)2 34. -3, 2, -22 2 i, 22
2 i; f (x)=2(x+3)(x-2)ax+ 22
2 ib ax- 22 2 ib 35. Domain: 5x0x≠-3, x≠36: horizontal asymptote: y=0; vertical asymptotes: x=-3, x=3
36. Domain: 5xx≠ -26; horizontal asymptote: y=1; vertical asymptote: x=-2
37. Step 1: R(x)=2(x-3) x ; domain: 5x0x≠06 Step 2: R is in lowest terms Step 3: no y-intercept; x-intercept: 3 Step 4: R is in lowest terms;
vertical asymptote: x =0 Step 5: Horizontal asymptote: y=2;
not intersected Step 6:
Step 7:
38. Step 1: Domain: 5x0x≠0, x≠26 Step 2: H is in lowest terms
Step 3: no y-intercept; x-intercept: -2 Step 4: H is in lowest terms;
vertical asymptote: x=0, x=2 Step 5: Horizontal asymptote: y=0;
intersected at (-2, 0) Step 6:
Step 7:
39. Step 1: R(x)=(x+3)(x-2) (x-3)(x+2); domain: 5x0x≠ -2, x≠36 Step 2: R is in lowest terms
Step 3: y-intercept: 1; x-intercepts: -3, 2 Step 4: R is in lowest terms;
vertical asymptotes: x= -2, x=3 Step 5: Horizontal asymptote: y=1;
intersected at (0, 1) Step 6:
Step 7:
9
5 5
6
y 10
x 10
(1, 4)(3, 0) (2, 5)
y 2
x 0 4,1
2
5 5
5
5
y 5
x 5 (1, 3) (2, 0)
3, 1 15
3, 5 3 1,1
3
x 2 x 0
5
5
5
7
y
x 5 5
(2, 0) (0, 1)
(3, 0)
4, 7 3
,11 9 5 2 4, 3
7
, 9 11 5 2
y 1
x 2 x 3
40. Step 1: F(x)= x3 (x+2)(x -2); domain:5x0x≠ -2, x≠26 Step 2: F is in lowest terms Step 3: y-intercept: 0; x-intercept: 0 Step 4: F is in lowest terms; vertical
asymptotes: x= -2, x=2 Step 5: Oblique asymptote: y=x;
intersected at (0, 0)
10
6
10
6
y
4 x (0, 0) 10
3, 27 5 1, 1
3
3, 27 5
y x x 2
x 2 1, 1
3
Step 6: Step 7:
41. Step 1: Domain: 5x0x≠16 Step 2: R is in lowest terms Step 3: y-intercept: 0; x-intercept: 0 Step 4: R is in lowest terms;
vertical asymptote: x=1
Step 5: No oblique or horizontal asymptote
42. Step 1: G(x)= (x+2)(x-2) (x+1)(x-2); domain: 5x0x≠ -1, x≠26 Step 2: In lowest terms, G(x)=x+2
x+1 Step 3: y-intercept: 2; x-intercept: -2 Step 4: Vertical asymptote: x= -1;
hole at a2, 4 3b
Step 5: Horizontal asymptote: y=1, not intersected
60
4 5 4
y 40
5 x (0, 0)
(2, 32) 2, 32
9 ,1
2 1 2
x 1
6 4
6
8
y 5
x 5 (0, 2) (2, 0)
3, 5 4 2, 4 3, 1 3
2
x 1 y 1
3 2, 1
43. (a) 5-3, 26 (b) (-3, 2) h (2, q) (c) (- q, -3] h526 (d) f (x)=(x - 2)2(x+3)
44. (a) y=0.25 (b) x= -2, x=2 (c) (-3, -2) h (-1, 2) (d) (- q, -34 h (-2, -14 h (2, q) (e) R(x)= x2+4x+3 4x2-16 45. 5x0x6 -2 or -16x626; (- q, -2) h (-1, 2) 46. 5x-4…x… -1 or xÚ16; 3-4, -14h31, q)
47. 5x0x61 or x726; (- q, 1) h (2, q) 48. 5x01…x…2 or x736; 31, 24h (3, q)
49. 5x0x6 -4 or 26x64 or x766; (- q, -4) h (2, 4) h (6, q)
21 2 4 1 1
1 2 1 2 3
4 2 4 6
Step 6: Step 7:
Step 6: Step 7:
50. (a) A1r2 =2pr2+500 r (b) 223.22 cm2 (c) 257.08 cm2 (d)
A is smallest when r≈3.41 cm.
51. (a)
The relation appears to be cubic.
(b) P(t)=4.4926t3-45.5294t2+136.1209t+115.4667; ≈+928,000 (c)
1000
00 8
280
0190 7
280
0190 7
52. (a) Even (b) Positive (c) Even (d) The graph touches the x-axis at x=0, but does not cross it there. (e) 8 Chapter Test (page 404)
1. 2. (a) 3 (b) p q: {1
2, {1, {3 2, {5
2, {3, {5, {15
2, {15 (c) -5, -1
2, 3; g(x)=(x+5)(2x+1)(x-3) (d) y-intercept: -15; x-intercepts: -5, -1
2, 3 (e) Crosses at -5, -1
2, 3 (f) y=2x3 (g) (-3.15, 60.30), (1.48, -39.00) (h)
y 7
8 x (4, 1)
(2, 1) (3, 2)
y 60
x 5 (3, 0) (1.48, 39.00) (0, 15)
0 1,
2 (3.15, 60.30)(2, 45)
(5, 0)
3. 4, -5i, 5i 4. e1, 5- 261
6 , 5+ 261
6 f 5. Domain: 5x0x≠ -10, x≠46; asymptotes: x=-10, y=2 6. Domain: 5x0x≠-16; asymptotes: x= -1, y=x+1
7. 8. Answers may vary. One possibility is f (x)=x4-4x3-2x2 +20x.
9. Answers may vary. One possibility is r(x)= 2(x-9)(x-1) (x-4)(x-9).
10. f (0) =8; f (4)= -36; Since f (0)=870 and f (4)= -3660, the Intermediate Value Theorem guarantees that there is at least one real zero between 0 and 4. 11. 5x0x63 or x786; (- q, 3) h (8, q)
y 5
x 5 (0, 3)
(3, 0) (1, 0)
x 1 y x 1
Cumulative Review (page 404)
1. 226 2. 5x0x…0 or xÚ16; (- q, 04 or 31, q) 3. 5x0-16x646; (-1, 4) 4. f (x)= -3x+1 5. y=2x-1 6.
7. Not a functions; 3 has two images. 8. 50, 2, 46 9. ex`xÚ3 2f; c3
2, qb
0 1 1 4 y
5
x 5 (1, 4)
y 6
x 5 (3, 5)
y 10
x 10 (1, 1) (2, 8)
(2, 8) (1, 1)
10. Center: (-2, 1); radius: 3
11. x-intercepts:-3, 0, 3; y-intercept: 0; symmetric with respect to the origin 12. y= -2 3x+17
3 13. Not a function; it fails the Vertical Line Test. 14. (a) 22 (b) x2-5x-2 (c) -x2-5x+2
(d) 9x2+15x-2 (e) 2x+h+5 15. (a) 5x0x≠16 (b) No; (2,7) is on the graph. (c) 4; (3, 4) is on the graph.
(d) 7 4; a7
4, 9b is on the graph. (e) Rational
y 5
x 2 (1, 1) (2, 4)
(2, 1) (2, 2) (5, 1)
16. 17. y 8
x 8 (0, 7)
7, 0 3
y 6
x 3 x 1 (0, 1)
(1, 1)
1 2
2, 0
1 2
2, 0
18. 6; y=6x-1 19. (a) x-intercepts: -5, -1, 5; y-intercept: -3 (b) No symmetry (c) Neither (d) Increasing: (- q, -3] and [2, q); decreasing: [-3, 2]
(e) Local maximum value is 5 and occurs at x= -3.
(f ) Local minimum value is -6 and occurs at x=2. 20. Odd
0 1 2 3
21. (a) Domain: 5x0x 7 -36 or (-3, q) (b) x-intercept: -1
2; y-intercept: 1 (c)
(d) Range: 5y0y656 or (- q, 5)
y 5
x (2, 2)5 (0, 1)
(2, 5)
(3, 5) 1, 0
2
y 8
x 2 (0, 2) (1, 5) (2, 2)
22. 23. (a) ( f+g)(x)=x2-9x-6; domain: all real numbers (b) af
gb(x)=x2-5x+1
-4x-7 ; domain: ex`x≠ -7 4f 24. (a) R(x)= -1
10x2+150x (b) +14,000 (c) 750; +56,250 (d) +75