1. {(fg)(x)=f(x)g(x)=(x−1)(x2−1)=x3−x2−x+1(f−g)(x)=f(x)−g(x)=(x−1)−(x2−1)=x−x2
No, the functions are not the same.
2. A gravitational force is still a force, so a(G(r)) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F)) does not make sense.
3. f(g(1))=f(3)=3 and g(f(4))=g(1)=3
4. g(f(2))=g(5)=3
5. A. 8; B. 20
6. [−4,0)∪(0,∞)
7. Possible answer:
g(x)=4+x2
h(x)=3−x4
f=h∘g
Solutions to Odd-Numbered Exercises
1. Find the numbers that make the function in the denominator g equal to zero, and check for any other domain restrictions on f and g, such as an even-indexed root or zeros in the denominator.
3. Yes. Sample answer: Let f(x)=x+1 and g(x)=x−1. Then f(g(x))=f(x−1)=(x−1)+1=x and g(f(x))=g(x+1)=(x+1)−1=x. So f∘g=g∘f.
5. (f+g)(x)=2x+6, domain: (−∞,∞)(f−g)(x)=2x2+2x−6[/latex],domain:[latex](−∞,∞)(fg)(x)=−x4−2x3+6x2+12x[/latex],domain:[latex](−∞,∞)(gf)(x)=6−x2x2+2x[/latex],domain:[latex](−∞,−6)∪(−6,6)∪(6,∞)
7. (f+g)(x)=2x4x3+8x2+1, domain: (−∞,0)∪(0,∞)(f−g)(x)=2x4x3+8x2−1[/latex],domain:[latex](−∞,0)∪(0,∞)(fg)(x)=x+2[/latex],domain:[latex](−∞,0)∪(0,∞)(gf)(x)=4x3+8x2[/latex],domain:[latex](−∞,0)∪(0,∞)
9. (f+g)(x)=3x2+x−5, domain: [5,∞)(f−g)(x)=3x2−x−5[/latex],domain:[latex][5,∞)(fg)(x)=3x2x−5[/latex],domain:[latex][5,∞)(gf)(x)=x−53x2[/latex],domain:[latex](5,∞)
11. a. 3; b. f(g(x))=2(3x−5)2+1; c. f(g(x))=6x2−2; d. (g∘g)(x)=3(3x−5)−5=9x−20; e. (f∘f)(−2)=163
13. f(g(x))=x2+3+2,g(f(x))=x+4x+7
15. f(g(x))=3x3x+1=x3x+1,g(f(x))=x3x+1
17. (f∘g)(x)=x2+4−41=2x,(g∘f)(x)=2x−4
19. f(g(h(x)))=(x+31)2+1
21. a. (g∘f)(x)=−2−4x3; b. (−∞,21)
23. a. (0,2)∪(2,∞); b. (−∞,−2)∪(2,∞); c. (0,∞)
25. (1,∞)
27. sample: {f(x)=x3g(x)=x−5
29. sample: {f(x)=x4g(x)=(x+2)2
31. sample: {f(x)=3xg(x)=2x−31
33. sample: {f(x)=4xg(x)=x+53x−2
35. sample: f(x)=xg(x)=2x+6
37.sample: f(x)=3xg(x)=(x−1)
39. sample: f(x)=x3g(x)=x−21
41. sample: f(x)=xg(x)=3x+42x−1
43. 2
45. 5
47. 4
49. 0
51. 2
53. 1
55. 4
57. 4
59. 9
61. 4
63. 2
65. 3
67. 11
69. 0
71. 7
73. f(g(0))=27,g(f(0))=−94
75. f(g(0))=51,g(f(0))=5
77. 18x2+60x+51
79. g∘g(x)=9x+20
81. 2
83. (−∞,∞)
85. False
87. (f∘g)(6)=6 ; (g∘f)(6)=6
89. (f∘g)(11)=11,(g∘f)(11)=11
91. c. Solve A(m(t))=4.
93. A(t)=π(25t+2)2 and A(2)=π(254)2=2500π square inches
95. A(5)=π(2(5)+1)2=121π square units
97. a. N(T(t))=23(5t+1.5)2−56(5t+1.5)+1;
b. 3.38 hours
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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..