1. The graphs of f(x) and g(x) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.
2.
a)
b)
3. g(x)=−f(x)
x
-2
0
2
4
g(x)
−5
−10
−15
−20
h(x)=f(−x)
x
-2
0
2
4
h(x)
15
10
5
unknown
4. even
5.
x
2
4
6
8
g(x)
9
12
15
0
6. g(x)=3x−2
7. g(x)=f(31x) so using the square root function we get g(x)=31x
8.
9. g(x)=x−11+1
10. Notice: g(x)=f(−x) looks the same as f(x) .
Solution to Odd-Numbered Exercises
1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.
3. A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.
5. For a function f, substitute (−x) for (x) in f(x). Simplify. If the resulting function is the same as the original function, f(−x)=f(x), then the function is even. If the resulting function is the opposite of the original function, f(−x)=−f(x), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.
7. g(x)=∣x−1∣−3
9. g(x)=(x+4)21+2
11. The graph of f(x+43) is a horizontal shift to the left 43 units of the graph of f.
13. The graph of f(x−4) is a horizontal shift to the right 4 units of the graph of f.
15. The graph of f(x)+8 is a vertical shift up 8 units of the graph of f.
17. The graph of f(x)−7 is a vertical shift down 7 units of the graph of f.
19. The graph of f(x+4)−1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f.
21. decreasing on (−∞,−3) and increasing on (−3,∞)
23. decreasing on (0,∞)
25.
27.
29.
31. g(x)=f(x−1),h(x)=f(x)+1
33. f(x)=∣x−3∣−2
35. f(x)=x+3−1
37. f(x)=(x−2)2
39. f(x)=∣x+3∣−2
41. f(x)=−x
43. f(x)=−(x+1)2+2
45. f(x)=−x+1
47. even
49. odd
51. even
53. The graph of g is a vertical reflection (across the x -axis) of the graph of f.
55. The graph of g is a vertical stretch by a factor of 4 of the graph of f.
57. The graph of g is a horizontal compression by a factor of 51 of the graph of f.
59. The graph of g is a horizontal stretch by a factor of 3 of the graph of f.
61. The graph of g is a horizontal reflection across the y -axis and a vertical stretch by a factor of 3 of the graph of f.
63. g(x)=∣−4x∣
65. g(x)=3(x+2)21−3
67. g(x)=21(x−5)2+1
69. The graph of the function f(x)=x2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.
71. The graph of f(x)=∣x∣ is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.
73. The graph of the function f(x)=x3 is compressed vertically by a factor of 21.
75. The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.
77. The graph of f(x)=x is shifted right 4 units and then reflected across the vertical line x=4.
79.
81.
Licenses & Attributions
CC licensed content, Shared previously
Precalculus.Provided by: OpenStaxAuthored by: Jay Abramson, et al..Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions.License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..