32 Chapter 1 Preliminaries

45. 46.

47. 48.

49. (a) domain: [0 2]; range: [ ] (b) domain: [0 2]; range: [ 1 0] # $

(c) domain: [0 2]; range: [0 2] (d) domain: [0 2]; range: [ 1 0]

(e) domain: [ 2 0]; range: [ 1] (f) domain: [1 3]; range: [ ] ! ! "

Section 1.5 Combining Functions; Shifting and Scaling Graphs 33

(g) domain: [ 2 0]; range: [ ] (h) domain: [ 1 1]; range: [ ] ! " ! "

50. (a) domain: [0 4]; range: [ 3 0] (b) domain: [ 4 0]; range: [ ] ! $

(c) domain: [ 4 0]; range: [ ] (d) domain: [ 4 0]; range: [ ] ! $ " %

(e) domain: [ 4]; range: [ 3 0] (f) domain: [ 2 2]; range: [ 3 0]#

34 Chapter 1 Preliminaries

(g) domain: [ 5]; range: [ 3 0] (h) domain: [0 4]; range: [0 3]"

51. y 3x 3 #

52. y 2x 1 x 1 % a b# #

53. y " " " " "# # # x x# #

54. y 1 1 " *$a bx x# #

55. y x 1 %

56. y 3 x 1

57. y 16 x % x# ## " #

58. y x % "$ #

59. y 3x 27x " " a b$ $

60. y " " x x# )$ $

61. Let y x f x and let g x x , h x x , i x x , and # " # a b a b a b a b "# " "# #"# "#

j x x f . The graph of h x is the graph of g x shifted left unit; the graph of i x is the grapha b a b a b a b a b # B" "# #"#

of h x stretched vertically by a factor of ; and the graph of j x f x is the graph of i x reflected across the x-axis.a b a b a b a b#

Section 1.5 Combining Functions; Shifting and Scaling Graphs 35

62. Let y f x Let g x x , h x x , and i x x f x " # # " a b a b a b a b a b a b a b a bx x# #"# "# "#"# The graph of g x is the graph of y x reflected across the x-axis. The graph of h x is the graph of g x shifted righta b a b a b two units. And the graph of i x is the graph of h x compressed vertically by a factor of .a b a b #

63. y f x x . Shift f x one unit right followed by a shift two units up to get g x x . " #a b a b a b a b$ 3

64. y x f x . Let g x x , h x x , i x x , and " B # " # " " #a b a b a b a b a b a b a b a b a b a b$ $ $ $$ j x x . The graph of h x is the graph of g x shifted right one unit; the graph of i x is the graph ofa b a b a b a b a b a b " # $ h x shifted down two units; and the graph of f x is the graph of i x reflected across the x-axis.a b a b a b

36 Chapter 1 Preliminaries

65. Compress the graph of f x horizontally by a factor of 2 to get g x . Then shift g x vertically down 1 unit toa b a b a b " "#x x get h x .a b ""#x

66. Let f x and g x Since , we see that the graph ofa b a b " " " " # "%" # " " " # " # B

x xx# #

B

#

#

# #

f x stretched horizontally by a factor of 1.4 and shifted up 1 unit is the graph of g x .a b a b

67. Reflect the graph of y f x x across the x-axis to get g x x. a b a b $ $

68. y f x x x x x . So the graph of f x is the graph of g x x # " # " # # a b a b a ba b a b a b a b a b a b#$ #$ #$ #$#$ #$ compressed horizontally by a factor of 2.