Algebra-Intermedia 2-páginas-68

Vista previa del material en texto

R28 Respuestas
83. 85. a) b) 5 87. a) b) 89. 30 mph 91. 5, 7
93. 5 pies por 12 pies 95. pies por 14.49 pies 97. 99. 101. 4%
103. 105. a) b)
c) 107. 2 108. 109. 110. 0 111.
Conjunto de ejercicios 8.2 1. 3. 5. Sí; si multiplica ambos lados de una
ecuación por �1 obtiene la otra ecuación. 7. a) b) c) Las respuestas variarán. 9. Dos soluciones reales. 11. No hay 
solución real 13. Dos soluciones reales 15. No hay solución real 17. Una solución real 19. Una solución real 21. 3, 6 23. 2, 4
25. 27. 29. 31. 33. 0, 5 35. 37. 39. 41. 43.
45. 47. 49. 51. 53. 55. 57. 59. 61.
63. 65. 0, 2 67. 69. 71. No hay número real 73.
75. 77. 79. 81. 83. 85.
87. a) b) 50 89. a) b) 15 91. Las respuestas variarán. 93. Sí 95. 3
97. w � 3 pies, l � 8 pies 99. 2 pulgadas 101. L 4.39 segundos 103. a) L 4.57 segundos b) L 4.79 segundos 105.
107. 109. a) L 1.94 segundos b) L 2.74 segundos c) La de Courtney
d) Sí, a los 1.5 segundos 110. 111. 7 112. 113. 114. No hay solución real
Conjunto de ejercicios 8.3 1. Las respuestas variarán. 3. 5. 7. 9. 11.
13. 15. 17. 19. 21.
23. 25. 27. 29. a) $10,950 b) 31. a) 32°F b) 80.8°F c) L2.92 minutos
33. a) 0.53 mil millones b) 2007 35. a) 111.4 mil millones de toneladas b) 2003 37. a) 1.301 millones b) 2009
39. l � 30 metros, w � 20 metros 41. 4 pies por hora 43. De ida a 6 mph, de regreso a 8 mph 45. Bonita L11.52 horas; Pamela L12.52 horas
47. 130 mph 49. Chris L11.76 horas; John L12.26 horas 51. 75 mph 53. l L 34.86 pulgadas, h L 19.61 pulgadas 55. Las respuestas variarán.
57. 6 metros por 3 metros o 2 metros por 9 metros 59. 60. 61. 62. 63. No hay solución.
Examen de mitad de capítulo 1. [8.1] 2. [8.1] 3. [8.1] 4. [8.1] 5. [8.1]
6. [8.1] 7. metros [8.1] 8. a) b) Dos soluciones reales distintas: b2 � 4ac � 0; una solución real:
b2 � 4ac � 0; no tiene soluciones reales: b2 � 4ac � 0 [8.2] 9. Dos soluciones reales y distintas [8.2] 10. [8.2] 11. [8.2]
12. [8.2] 13. [8.2] 14. [8.2] 15. 10 lámparas [8.2] 16. [8.3]
17. [8.3] 18. [8.3] 19. 5 pies por 12 pies [8.3] 20. 5 relojes [8.3]
Conjunto de ejercicios 8.4 1. Puede escribirse en la forma au2 � bu � c � 0. 3. u � x2; da la ecuación 3u2 � 5u � 1 � 0.
5. u � z�1; da la ecuación u2 � u � 56. 7. 9. 11. 13. 15. 17.
19. 21. 23. 4 25. 9 27. 29. 31. 33. 35. 37.
39. 41. 3, 4 43. 45. 47. 49. 1, 27 51. 27, 216 53. 55. 57. (1, 0), (16, 0)
59. Ninguna 61. 63. 65. 67. 69. Sea u � x2
71. Sea u � x�1 73.
75. 77. No; las soluciones imaginarias siempre ocurren en 
parejas. 79. a) y b) 81. 83. 85. 2, 1 87. 89. 91. 92. 1
93. 94. 95. 9132xy213 2D: �, R: {y ƒy Ú 0}
43
60
;B3 ; 115
2
-3, 1, 2, -42, 
1
4
- 
14
5
, - 
8
3
1
5
, - 
1
4
x4
- 7x2
+ 10 = 0; inicie con 1x + 1221x - 1221x + 1521x - 152 = 0
x4
- 5x2
+ 4 = 0; inicie con 1x - 221x + 221x - 121x + 12 = 0
1;2, 02, 1;5, 021-1, 02, 14, 021-8, 02, 127, 021-4, 02, a1
5
, 0b
-32, -1
1
4
- 
1
2
, 
1
6
1, - 
1
10
2, 
1
3
- 
1
2
; 
516
6
, ; 
139
3
-6, - 
5
2
;16, ;1
4
3
, - 
1
2
1, -9
1
9
;1, ; i15;3, ; i12
;13, ;15; 
1
2
, ;2;2, ;13;2, ;3; i, ;4i;1, ;3
y = 2D2
- x2x = A3A
k
r = 2x2
- yx2
- 4x - 1 = 0x2
- 5x - 14 = 0
1 ; i114
3
-2 ; 213- 
5
3
, 
3
2
b2
- 4ac16 + 6122-1 ; i1143
8
2 ; 114-6, 2- 
1
2
, - 
13
2
3 ; 2i15;712
x2
y32
16
r - 4
R =
E - Ir
I
-16
L7c = 21v¿22 + v2v1 = 2v2
2
- 2adv = A2E
m
t = Ah - s0
-16
 o t =
1s0 - h
4
H = 2d2
- L2
- W2b = 2c2
- a2W = 2d2
- L2r = A3V
ph
c = AE
m
t =
1d
4
i = AE
r
t = A d
4.9
S = 1A
6y - x
3xy
12, -125.0 * 102 o 500
1- 0.12 + 114.39522>1.2 L 3.0618 milímetros
215, -15
n150 - 0.4n2 = 660n110 - 0.02n2 = 450
x2
- 4x + 13 = 0x2
- 6x + 7 = 0x2
+ 9 = 0x2
- 2 = 015x2
- x - 6 = 0x2
+ 8x - 9 = 0
x2
- 7x + 10 = 0
7 ; 117
4
-5, 6
-0.6 ; 10.84
0.2
 o -3 ; 121
-1 ; i123
4
2 ; i16
2
3 ; 133
2
3 ; 1309
30
-6 ; 216
3
1
3
, - 
2
3
4, -6
1
2
, - 
5
3
-3, 
1
2
-3 ; i115
2
1 ; 12
1
4
-1
2 ; i12
2
2 ; i111
3
;8-2 ; 2161, -7
-84b2
- 4ac
a = -3, b = 6, c = 8x =
-b ; 2b2
- 4ac
2a
4x3
+ x2
- 21x + 6e10, 
4
3
f$4200 al 7%, $5800 al 6 
1
4
%r = - 5 + A80 + 25p
p
L 2.1 pulgadas
r =
41p
p
L 2.26 pulgadasS = 32 + 801p L 173.80 pulgadas cuadradasL6%
124 L 4.90 pies1200 L 14.14 pulgadas
12 + 1288
2
L 14.49
-3 + 11918 = 1x + 421x + 2221 = 1x + 221x - 221 ; i
Respuestas R29
Conjunto de ejercicios 8.5 1. La gráfica de una ecuación cuadrática se denomina parábola. 3. El eje de simetría de una
parábola es la recta donde, si la gráfica se dobla, los dos lados quedan uno sobre el otro. 5. 7. a) Cuando a � 0, f(x)
tendrá un mínimo ya que la gráfica abre hacia arriba. b) Cuando a � 0, f(x) tendrá un máximo ya que la gráfica abre hacia abajo.
9. Haga x � 0 y resuelva para y.
¢ - 
b
2a
, 
4ac - b2
4a
≤
11. a) b)
13. Valor mínimo; la gráfica abre hacia
arriba
15. a) Hacia arriba b) (0, 15)
c) d)
e)
17. a) Hacia arriba b) (0, 3)
c) d) (1, 0), (3, 0)
e)
x7�3
y
5
�3
7
f(x) � x2 � 4x � 3
12, -12
x3�7
y
5
�5
f(x) � x2 � 8x � 15
1-5, 02, 1-3, 021-4, -12
19. a) Hacia abajo b) (0, 8)
c) d) (2, 0)
e)
21. a) Hacia abajo b) (0, 5)
c) (2, 9) d) (5, 0)
e)
23. a) Hacia abajo b)
c) d) No hay intercepción
con el eje x
e)
x7�3
y
3
�7
t(x) � �x2 � 4x � 5
12, -12
10, -52
x7�3
y
9
g(x) � �x2 � 4x � 5
1-1, 02,
x5�5
y
9
f(x) � �x2 � 2x � 8
1-4, 02,1-1, 92
25. a) Hacia arriba b) (0, 4)
c) (2, 0) d) (2, 0)
e)
27. a) Hacia arriba b) (0, 2)
c) (0, 2) d) No hay intersecciones con
el eje x
e)
29. a) Hacia abajo b) (0, 5)
c) (0, 5) d)
e)
x5�5
y
5
�5
l(x) � �x2 � 5
1-15, 02, 115, 02
x5�5
y
9
r(x) � x2 � 2
x7�3
y
7
�3
f(x) � x2 � 4x � 4
31. a) Hacia abajo b)
c) d) No hay intersecciones
con el eje x
e)
33. a) Hacia arriba b) (0, 3)
c) d) No hay intersecciones
con el eje x
e)
35. a) Hacia arriba b)
c)
d)
e)
x5�5
y
�9
y � 3x2 � 4x � 6
a -2 + 122
3
, 0b , a -2 - 122
3
, 0b
a - 
2
3
, - 
22
3
b
10, -62
x5�5
y
9
m(x) � 3x2 � 4x � 3
a - 
2
3
, 
5
3
b
x9
y
�9
f(x) � �2x2 � 4x � 8
11, -62
10, -82
37. a) Hacia arriba b)
c) d) (2, 0)
e)
39. a) Hacia abajo b)
c)
d) No hay intersecciones con el eje x.
e)
41.
x9
y
5
�5
f(x) � (x � 3)2
(3, 0)
x9
y
�5 f(x) � �x2 � 3x � 5
a3
2
, - 
11
4
b
10, -52
x5�5
y
3
�7 y � 2x2 � x � 6
a - 
3
2
, 0b ,a1
4
, - 
49
8
b
10, -62
43. 45. 47. 49. 51.
x�9
y
9
(�4, 4)
f(x) � (x � 4)2 � 4
x9
y
9
(2, 3)
f(x) � (x � 2)2 � 3
x�5
y
9
(0, �1)
f(x) � x2 � 1
x5�5
y
9
(0, 3)f(x) � x2 � 3
x3�7
y
7
�3
f(x) � (x � 1)2
(�1, 0)
R30 Respuestas
53. 55. 57.
x5�5
y
3
�7
h(x) � �2(x � 1)2 � 3
(�1, �3)x7�3
y
5
�5
(2, 2)
y � �2(x � 2)2 � 2
x3�7
y
3
�7
g(x) � �(x � 3)2 � 2
(�3, �2)
59. a)
b)
61. a)
b)
x5�5
y
5
�5
g(x) � x2 � x � 3
1 , �
2
13
4
g1x2 = ax -
1
2
b 2
-
13
4
x9
y
5
�5
(3, �1)
f(x) � x2 � 6x � 8
f1x2 = 1x - 322 - 1
63. a)
b)
65. a)
b)
67. a)
b)
x5�5
y
3
�7
f(x) � 2x2 � 5x � 3
5 , ��
4
49
8
f1x2 = 2ax +
5
4
b 2
-
49
8
x7�3
y
5
�9
g(x) � x2 � 4x � 1
(2, �5)
g1x2 = 1x - 222 - 5
x5�5
y
�9
f(x) � �x2 � 4x � 6
(�2, �2)
f1x2 = -1x + 222 - 2
69. d) 71. b) 73. a) b) 75. a) b)
77. a) b) 79. 2010 81. 4 unidades 83. 3 unidades 85. 87.
89. a) Las gráficas tendrán las mismas intersecciones con el eje x, pero f(x) � x2 � 8x � 12 abrirá hacia arriba y g(x) � �x2 � 8x � 12 
abrirá hacia abajo. b) Sí, ambas en (6, 0) y (2, 0) c) No; vértice de f(x) en (4, �4), vértice de g(x) en (4, 4)
d) 91. a) b) $2 c) $22 d) $12 e) $10,000 93. a) 100 b) $3800
95. a) 40.425 metros b) 2.5 segundos c) L5.37 segundos
97. a) b) 2000 99. 400 pies cuadrados
101. 4 y 103. 900, 30 y 30 105. a) $142,400 b) 380
107. a) f(t) � �16(t � 1.625)2 � 45.25 b) 45.25 pies, 1.625 segundos c) Iguales
109. 200p pies cuadrados 110. 111. 112. 113.
Conjunto de ejercicios 8.6 1. a) o b) 3. Sí; 5. Sí, �2 y 1 hace que la fracción sea 0; no, �1
hace que la fracción sea indefinida. 7. 9. 11. 13. 15.
17. 19. 21. 23. 25.
27. 29. 31. 33.
35. 37. 39. 41. 43. 45.
47. 49. 51. 53. 55.
57. 59. 61. 63. 65.
67. 69. 71. 73. 75. 77.
79. 81. 83. 85. a) (4, q); y � 0en este intervalo b) (�q, 2) ´ (2, 4); y � 0 en este
intervalo 87. 89. 91. Todos los números reales; para cualquier valor de x, la expresión es � 0.
93. Todos los números reales excepto �2; para cualquier valor de x excepto �2, la expresión es � 0. 95. No hay solución; la gráfica abre
x + 3
x - 4
Ú 0x2
+ 2x - 8 7 0
2
3
4
7
0�241
3�21�264
c - 
5
2
, 3 d ´ 14, q21- q , -42 ´ 11, 6]1- q , -82 ´ [0, 32
1-2, 1] ´ [7, q21-3, 22 ´ 15, q21- q , -32 ´ 1-1, 62ex ` - 
8
3
… x 6 2 fex ` - 
4
3
6 x 6
1
2
f
ea ` a … -2 o a 7
1
3
f{y ƒ -4 6 y … -2}{c ƒc 6 4 o c 7 10}{a ƒ -5 6 a 6 9}{x ƒx … -3 o x 7 2}
{x ƒ -5 6 x 6 1}{x ƒx 6 -2 o x 7 4}
5
2
0�7
4
129�9 � 
4
129�9 � 860�4
60
1- q , 02[-2, -2] ´ c8
3
, q ba -1, - 
5
3
b ´ 13, q2
a -6, - 
5
2
b ´ 12, q21- q , -42 ´ 1-2, 32[-5, -1] ´ [2, q2
2
26 � 3
2
26 � 34
5�2
1
2�3
4�40�6 �1�2 4
Ú2 6 x 6 5x 7 5x 6 2
x
x + 6
-81-2, 3, 22
x5�5
y
5
�5
-4-16,
L$577
x27
y
140
60
I � �x2 � 24x � 44, 0 � x � 24
x10�10
y
10
�10
f(x) � x2 � 8x � 12
g(x) � �x2 � 8x � 12
f1x2 = -4ax +
3
5
b 2
- 12f1x2 = 21x - 322 - 2R = $800n = 200
A = 240.25x = 10.5A = 121x = 7
Respuestas R31
hacia arriba y no tiene intersecciones con el eje x, de modo que siempre está por arriba del eje x. 97. 99. x2 � 3x � 0;
multiplique los factores que tienen valores frontera. 101. x2 � 0; x2 siempre es � 0. 103.
105. 109. 6 cuartos 110. 111. 112. 113.
Ejercicios de repaso del capítulo 8 1. 2. 3. 4. 5. 3, 4 6. 7.
8. 9. 10. 11. a) b) 3 12. a) b) 5 13. 6, 7
14. L16.90 pies por L16.90 pies 15. Dos soluciones reales 16. No hay solución real 17. Una solución real 18. No hay soluciones reales
19. Una solución real 20. Dos soluciones reales 21. 22. 2, 9 23. 24. 25. 26. 27.
28. 29. 30. 31. 32. 33. 34. 35. 36.
37. 38. 39. 40. 41. 42.
43. 8 pies por 12 pies 44. 45. 4% 46. 7, 11 47. 8 pulgadas por 12 pulgadas 48. $540 49. a) $121.8 mil millones
b) 2010 50. a) 720 pies b) 7 segundos 51. a) 40 mililitros b) 150°C 52. la mayor L23.51 horas; la menor L24.51 horas
53. 50 millas por hora 54. 1.6 millas por hora 55. l � 10 unidades, w � 8 unidades 56. 20 mesas 57. 58.
59. 60. 61. 62. 63. 64. 65.
66. 67. 68. 69. 70. 71. Ninguno 72. 13 ; 117, 02, 13 ; 16, 02a 4
25
, 0b1;1, 02, 1;9, 02- 
1
5
, - 
5
2
4, 
13
8
27
8
, 8
1
9
3
2
, - 
1
6
;212, ; i13;4, ;15;2, ;3v2 = 2v1
2
+ 2advy = 2v2
- vx
2
t = Ac - h
4.9
a = 2c2
- b2
1128 L 11.31
x2
- 6x + 13 = 0x2
- 11 = 03x2
+ 4x - 4 = 0x2
- 2x - 3 = 0
3 ; 313
2
7 ; 189
10
2
3
, - 
3
2
10, -6
1
4
, - 
3
2
5
2
, - 
5
3
1 ; i110
1 ; i151
2
3 ; 133
3
-2 ; 110
2
-2 ; 213
-4 ; 111
1
4
, -3
3
2
, - 
5
3
0, 
9
7
8, -50, - 
4
3
63 = 1x + 221x + 4232 = 1x + 121x + 522 ; 2i171 ; 3i-3 ; 121
-1 ; 1104, -8
5
4
, - 
3
4
- 
1
3
, 1
-1 ; 2115
2
5 ; 216
38 - 9i
x - 3
x + 1
3r + 3s - 9t- 
1
2
[-2, -1] ´ [2, q2
1- q , -32 ´ 1-1, 12 ´ 13, q23�8 �1�5
73. a) Hacia arriba b) (0, 0)
c) d)
e)
74. a) Hacia arriba b)
c) d)
e)
75. a) Hacia abajo b)
c) d) No hay intersecciones
con el eje x
e)
x5�5
y
3
�7
g(x) � �x2 � 2
(0, �2)
10, -22
10, -22
x5�5
y
�9
f(x) � x2 � 2x � 8
(1, �9)
1-2, 02, 14, 0211, -92
10, -82
x3�7
y
3
�7
f(x) � x2 � 5x
5 , ��
2
25
4
10, 02, 1-5, 02a - 
5
2
, - 
25
4
b
76. a) Hacia abajo b) (0, 15)
c) d)
e)
77. a) $11 b) $7600
78. a) 2.5 segundos b) 175 pies
79.
80.
x3�7
y
�9
f(x) � �(x � 2)2 � 3
(�2, �3)
x7�3
y
7
�3
f(x) � (x � 3)2
(3, 0)x5�5
y
�6
g(x) � �2x2 � x � 15
1 , �
4
121
8
1-3, 02, a5
2
, 0ba - 
1
4
, 
121
8
b
81. 82. 83. 84. 85. 86.
87. 88.
89. 90. 91.
92. 93. 94. 95.
96. 97. 98. 99. 100.
101. 102. 103.
Examen de práctica del capítulo 8 1. [8.1] 2. [8.1] 3. [8.2] 4. [8.2] 5. [8.1–8.2]
6. [8.1–8.2] 7. [8.2] 8. [8.3] 9. a) $121,200 b) L$2712.57 pies cuadrados [8.1–8.3]v = A2K
m
5x2
- 18x - 8 = 0
-7 ; 133
2
2
3
, -12 ; i178, -23 ; 123, -5
23
10
5
3
2�2�4�9
1- q , -32 ´ [0, 6]1-2, 3] ´ 17, q21- q , -32 ´ 12, 821-2, 02 ´ 14, q21- q , -42 ´ 1-2, 02
c - 
4
3
, 1 d ´ [3, q2{x ƒx … 0 o 3 … x … 6}{x ƒ -4 6 x 6 -1 o x 7 2}ex ` - 
5
3
6 x 6 6 f
{x ƒx 6 -3 o x Ú 2}{x ƒ -2 6 x … 3}{x ƒx 6 -1 o x 7 5}
5� 53
2
3
2
�
4
3�4
2
4111 � 
2
4111 � 2�5�1�3
x5�5
y
9
h(x) � (x � 1)2 � 31
2
(1, 3)
x3
y
�9
g(x) � �2(x � 4)2 � 1
(�4, �1)
R32 Respuestas
10. 50 mph [8.1–8.3] 11. [8.4] 12. [8.4] 13. [8.4] 14. [8.5]
15. [8.5] 16. Dos soluciones reales [8.5] 17. a) Hacia arriba b)
c) d) e) [8.5]
18. [8.5] 19. [8.6] 20. [8.6] 21. a) b) [8.6]
22. w � 5 pies, l � 13 pies [8.5] 23. 6 segundos [8.5] 24. a) 20 b) $ 490 [8.5] 25. 30 [8.5]
Examen de repaso acumulativo 1. 13 [1.4] 2. 18 [1.4] 3. [1.6] 4. [2.6] 5. [2.1]
6. Todos los números reales, [2.1] 7. [2.5] 8. [3.4] 9. 1500 [3.2] 10. [3.5]
11. a) No, las gráficas no pasan la prueba de la recta vertical b) Dominio: Rango: [3.2]
12. a) b) [3.3] 13. 160 [4.5] 14. [4.1] 15. [5.5]
16. a) b) [5.7] 17. [6.2] 18. [6.4]
19. 11.52 watts [6.6] 20. [7.7]
Capítulo 9
Conjunto de ejercicios 9.1 1. Para determinar sustituya g(x) por x en f(x). 3. a) Cada y tiene una única.
b) Utilice la prueba de la recta horizontal. 5. a) Sí; cada primer coordenada está asociada con una sola segunda coordenada. b) Si; cada
segunda coordenada está asociada con sólo una primer coordenada. c) invertir la pareja ordenada.
7. El dominio de f es el rango de f�1 y el rango de f es el dominio de f�1. 9. a) b) 37 c) d) 19 11. a)
b) 19 c) d) 52 13. a) b) c) d) 15. a) b) c) d)
17. a) b) 442 c) d) 294 19. a) b) c) d) 21. No 23. Sí
25. Sí 27. No 29. Sí 31. No 33. No 35. Sí 37. Sí 39. No 41. Sí 43. f(x): Dominio: 5�2, �1, 2, 4, 86; rango: 50, 4, 6, 7, 96;
f�1(x): Dominio: 50, 4, 6, 7, 96; Rango: 5�2, �1, 2, 4, 86 45. f(x): Dominio: 5�1, 1, 2, 46; Rango: 5�3, �1, 0, 26, f�1(x): Dominio: 5�3, �1, 0, 26;
Rango: 5�1, 1, 2, 46 47. f(x):
49. a) Sí b) 51. a) Sí b) 53. a) No 55. a) No 57. a) Sí b) 59. a) No
61. a) Sí b) 63. a) Sí b) 65. a) Sí b) h-11x2 = 1x + 4, x Ú -4g-11x2 = x2
- 2, x Ú 0g-11x2 = 13 x + 6
g-11x2 =
1
x
h-11x2 =
x
4
f-11x2 = x + 2
Rango: {y ƒy Ú 2}Dominio: {x ƒx Ú 0};Rango: {y ƒy Ú 0}; f-11x2;Dominio: {x ƒx Ú 2};
151x + 1-11x + 5 - 4x4
+ 2x2
+ 6x4
+ 10x2
+ 26
3
13
3
3x + 1
3 
1
4
9
x
+ 13 
1
2
2
x
+ 3
1
11
1
2x + 3
x2
+ 7x + 8
x2
+ x - 1x2
+ 3x2
+ 4x + 5
{15, 32, 12, 42, 13, -12, 1-2, 02};
1f � g21x2,
-14 - 23i
29
12
5
21x - 42
1x - 321x - 521a + b22a2
+ 2ab + b2
1x + 921x + 72a5
2
, 0b
x5�5
y
5
�5
y � 2
x5�5
y
5
�5
x � �4
�{x ƒx Ú -2},
y = x - 1m = - 
9
7
, a0, 
15
7
b1-12, 82�
3x - 7e - 
1
2
, 
9
2
f2.54 * 106
ex ` - 
5
2
… x 6 -2 fc - 
5
2
, -2b
4�1�57�6
2x2
+ 13x - 7 = 0
x5�5
y
�9
y � x2 � 2x � 8
1-4, 02, 12, 021-1, -92
10, -82
x7�3
y
�9
(2, �2)
h(x) � � (x � 2)2 � 21
2
x7�3
y
9
f(x) � (x � 3)2 � 2
(3, 2)
a 9
16
, 0b343
27
, -216; 
110
2
, ; i110
67. a)
b)
x10�10
y
10
�10
f�1(x)
f(x)
f-11x2 =
x - 8
2
69. a)
b)
x9
y
9
f(x)
f�1(x)
f-11x2 = x2, x Ú 0 71. a)
b)
x5�5
y
5
�5
f�1(x)
f(x)
f-11x2 = x2
+ 1, x Ú 0 73. a)
b)
x10
�10
y
10
�10
f�1(x)
f(x)
f-11x2 = x3
75. a)
b)
x5�5
y
5
�5
f�1(x)
f(x)
f-11x2 =
1
x
, x 7 0 77. 79.
81. 83.
85. No, la composición de funciones no es conmutativa. Sea y 
Entonces mientras que
87. a) b) El dominio es , para todas ellas.
89. El rango de es el dominio de 91. está en pies y f�1(x) está en yardas.
93. 95. x está en libras, está en gramos.
97. x está en yardas, está en metros. 99. Sí 101. Sí1f � g21x21f � g21x2 = 0.915x,
1f � g21x21f � g21x2 = 453.6x,f-11x2 =
9
5
 x + 32.
f-11x2 =
x
3
; xf1x2.f-11x2
�1f � g21x2 = x; 1g � f21x2 = x
1g � f21x2 = x2
+ 1.1f � g21x2 = x2
+ 2x + 1,
g1x2 = x + 1.f1x2 = x2
1f � f-121x2 = x, 1f-1 � f21x2 = x1f � f-121x2 = x, 1f-1 � f21x2 = x
1f � f-121x2 = x, 1f-1 � f21x2 = x1f � f-121x2 = x, 1f-1 � f21x2 = x
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 /ENU <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>
 >>
>> setdistillerparams
<<
 /HWResolution [2400 2400]
 /PageSize [612.000 792.000]
>> setpagedevice
<<
 /ASCII85EncodePages false
 /AllowTransparency false
 /AutoPositionEPSFiles true
 /AutoRotatePages /None
 /Binding /Left
 /CalGrayProfile (Gray Gamma 1.8)
 /CalRGBProfile (Apple RGB)
 /CalCMYKProfile (U.S. Sheetfed Uncoated v2)
 /sRGBProfile (sRGB IEC61966-2.1)
 /CannotEmbedFontPolicy /Warning
 /CompatibilityLevel 1.4
 /CompressObjects /Off
 /CompressPages true
 /ConvertImagesToIndexed true
 /PassThroughJPEGImages true
 /CreateJDFFile false
 /CreateJobTicket false
 /DefaultRenderingIntent /Default
 /DetectBlends true
 /DetectCurves 0.1000
 /ColorConversionStrategy /LeaveColorUnchanged
 /DoThumbnails false
 /EmbedAllFonts true
 /EmbedOpenType false
 /ParseICCProfilesInComments true
 /EmbedJobOptions true
 /DSCReportingLevel 0
 /EmitDSCWarnings false
 /EndPage -1
 /ImageMemory 1048576/LockDistillerParams false
 /MaxSubsetPct 100
 /Optimize true
 /OPM 1
 /ParseDSCComments true
 /ParseDSCCommentsForDocInfo true
 /PreserveCopyPage true
 /PreserveDICMYKValues true
 /PreserveEPSInfo true
 /PreserveFlatness true
 /PreserveHalftoneInfo false
 /PreserveOPIComments false
 /PreserveOverprintSettings true
 /StartPage 1
 /SubsetFonts true
 /TransferFunctionInfo /Remove
 /UCRandBGInfo /Preserve
 /UsePrologue false
 /ColorSettingsFile (Color Management Off)
 /AlwaysEmbed [ true
 ]
 /NeverEmbed [ true
 ]
 /AntiAliasColorImages false
 /CropColorImages true
 /ColorImageMinResolution 150
 /ColorImageMinResolutionPolicy /OK
 /DownsampleColorImages true
 /ColorImageDownsampleType /Bicubic
 /ColorImageResolution 300
 /ColorImageDepth -1
 /ColorImageMinDownsampleDepth 1
 /ColorImageDownsampleThreshold 1.50000
 /EncodeColorImages true
 /ColorImageFilter /DCTEncode
 /AutoFilterColorImages true
 /ColorImageAutoFilterStrategy /JPEG
 /ColorACSImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /ColorImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /JPEG2000ColorACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /JPEG2000ColorImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /AntiAliasGrayImages false
 /CropGrayImages true
 /GrayImageMinResolution 150
 /GrayImageMinResolutionPolicy /OK
 /DownsampleGrayImages true
 /GrayImageDownsampleType /Bicubic
 /GrayImageResolution 300
 /GrayImageDepth -1
 /GrayImageMinDownsampleDepth 2
 /GrayImageDownsampleThreshold 1.50000
 /EncodeGrayImages true
 /GrayImageFilter /DCTEncode
 /AutoFilterGrayImages true
 /GrayImageAutoFilterStrategy /JPEG
 /GrayACSImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /GrayImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /JPEG2000GrayACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /JPEG2000GrayImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /AntiAliasMonoImages false
 /CropMonoImages true
 /MonoImageMinResolution 1200
 /MonoImageMinResolutionPolicy /OK
 /DownsampleMonoImages true
 /MonoImageDownsampleType /Bicubic
 /MonoImageResolution 1200
 /MonoImageDepth -1
 /MonoImageDownsampleThreshold 1.50000
 /EncodeMonoImages true
 /MonoImageFilter /FlateEncode
 /MonoImageDict <<
 /K -1
 >>
 /AllowPSXObjects false
 /CheckCompliance [
 /None
 ]
 /PDFX1aCheck false
 /PDFX3Check false
 /PDFXCompliantPDFOnly false
 /PDFXNoTrimBoxError true
 /PDFXTrimBoxToMediaBoxOffset [
 0.00000
 0.00000
 0.00000
 0.00000
 ]
 /PDFXSetBleedBoxToMediaBox true
 /PDFXBleedBoxToTrimBoxOffset [
 0.00000
 0.00000
 0.00000
 0.00000
 ]
 /PDFXOutputIntentProfile (None)
 /PDFXOutputConditionIdentifier ()
 /PDFXOutputCondition ()
 /PDFXRegistryName (http://www.color.org)
 /PDFXTrapped /Unknown
 /Description <<
 /JPN <FEFF3053306e8a2d5b9a306f30019ad889e350cf5ea6753b50cf3092542b308030d730ea30d730ec30b9537052377528306e00200050004400460020658766f830924f5c62103059308b3068304d306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103057305f00200050004400460020658766f8306f0020004100630072006f0062006100740020304a30883073002000520065006100640065007200200035002e003000204ee5964d30678868793a3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002>
 /FRA <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>
 /DEU <FEFF00560065007200770065006e00640065006e0020005300690065002000640069006500730065002000450069006e007300740065006c006c0075006e00670065006e0020007a0075006d002000450072007300740065006c006c0065006e00200076006f006e0020005000440046002d0044006f006b0075006d0065006e00740065006e0020006d00690074002000650069006e006500720020006800f60068006500720065006e002000420069006c0064006100750066006c00f600730075006e0067002c00200075006d002000650069006e00650020007100750061006c00690074006100740069007600200068006f006300680077006500720074006900670065002000410075007300670061006200650020006600fc0072002000640069006500200044007200750063006b0076006f0072007300740075006600650020007a0075002000650072007a00690065006c0065006e002e00200044006900650020005000440046002d0044006f006b0075006d0065006e007400650020006b00f6006e006e0065006e0020006d006900740020004100630072006f0062006100740020006f0064006500720020006d00690074002000640065006d002000520065006100640065007200200035002e003000200075006e00640020006800f600680065007200200067006500f600660066006e00650074002000770065007200640065006e002e00200042006500690020006400690065007300650072002000450069006e007300740065006c006c0075006e00670020006900730074002000650069006e00650020005300630068007200690066007400650069006e00620065007400740075006e00670020006500720066006f0072006400650072006c006900630068002e>
 /PTB <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>
 /DAN <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>
 /NLD <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>/ESP <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>
 /SUO <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>
 /ITA <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>
 /NOR <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>
 /SVE <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>
 /ENU <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>
 >>
>> setdistillerparams
<<
 /HWResolution [2400 2400]
 /PageSize [612.000 792.000]
>> setpagedevice
<<
 /ASCII85EncodePages false
 /AllowTransparency false
 /AutoPositionEPSFiles true
 /AutoRotatePages /None
 /Binding /Left
 /CalGrayProfile (Gray Gamma 1.8)
 /CalRGBProfile (Apple RGB)
 /CalCMYKProfile (U.S. Sheetfed Uncoated v2)
 /sRGBProfile (sRGB IEC61966-2.1)
 /CannotEmbedFontPolicy /Warning
 /CompatibilityLevel 1.4
 /CompressObjects /Off
 /CompressPages true
 /ConvertImagesToIndexed true
 /PassThroughJPEGImages true
 /CreateJDFFile false
 /CreateJobTicket false
 /DefaultRenderingIntent /Default
 /DetectBlends true
 /DetectCurves 0.1000
 /ColorConversionStrategy /LeaveColorUnchanged
 /DoThumbnails false
 /EmbedAllFonts true
 /EmbedOpenType false
 /ParseICCProfilesInComments true
 /EmbedJobOptions true
 /DSCReportingLevel 0
 /EmitDSCWarnings false
 /EndPage -1
 /ImageMemory 1048576
 /LockDistillerParams false
 /MaxSubsetPct 100
 /Optimize true
 /OPM 1
 /ParseDSCComments true
 /ParseDSCCommentsForDocInfo true
 /PreserveCopyPage true
 /PreserveDICMYKValues true
 /PreserveEPSInfo true
 /PreserveFlatness true
 /PreserveHalftoneInfo false
 /PreserveOPIComments false
 /PreserveOverprintSettings true
 /StartPage 1
 /SubsetFonts true
 /TransferFunctionInfo /Remove
 /UCRandBGInfo /Preserve
 /UsePrologue false
 /ColorSettingsFile (Color Management Off)
 /AlwaysEmbed [ true
 ]
 /NeverEmbed [ true
 ]
 /AntiAliasColorImages false
 /CropColorImages true
 /ColorImageMinResolution 150
 /ColorImageMinResolutionPolicy /OK
 /DownsampleColorImages true
 /ColorImageDownsampleType /Bicubic
 /ColorImageResolution 300
 /ColorImageDepth -1
 /ColorImageMinDownsampleDepth 1
 /ColorImageDownsampleThreshold 1.50000
 /EncodeColorImages true
 /ColorImageFilter /DCTEncode
 /AutoFilterColorImages true
 /ColorImageAutoFilterStrategy /JPEG
 /ColorACSImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /ColorImageDict <<
 /QFactor0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /JPEG2000ColorACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /JPEG2000ColorImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /AntiAliasGrayImages false
 /CropGrayImages true
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 >>
>> setdistillerparams
<<
 /HWResolution [2400 2400]
 /PageSize [612.000 792.000]
>> setpagedevice