Solutions 4: Composition of Functions

Solutions for Try Its

1. 
{(fg)(x)=f(x)g(x)=(x1)(x21)=x3x2x+1(fg)(x)=f(x)g(x)=(x1)(x21)=xx2\begin{cases}\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)=\left(x - 1\right)\left({x}^{2}-1\right)={x}^{3}-{x}^{2}-x+1\\ \left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)=\left(x - 1\right)-\left({x}^{2}-1\right)=x-{x}^{2}\end{cases}


No, the functions are not the same.

2. A gravitational force is still a force, so
a(G(r))a\left(G\left(r\right)\right)
makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but
G(a(F))G\left(a\left(F\right)\right)
does not make sense.

3. 
f(g(1))=f(3)=3f\left(g\left(1\right)\right)=f\left(3\right)=3
and
g(f(4))=g(1)=3g\left(f\left(4\right)\right)=g\left(1\right)=3


4. 
g(f(2))=g(5)=3g\left(f\left(2\right)\right)=g\left(5\right)=3


5. A. 8; B. 20

6. 
[4,0)(0,)\left[-4,0\right)\cup \left(0,\infty \right)


7. Possible answer:

g(x)=4+x2g\left(x\right)=\sqrt{4+{x}^{2}}

h(x)=43xh\left(x\right)=\frac{4}{3-x}

f=hgf=h\circ g

 

Solutions to Odd-Numbered Exercises

1. Find the numbers that make the function in the denominator
gg
equal to zero, and check for any other domain restrictions on
ff
and
gg
, such as an even-indexed root or zeros in the denominator.

3. Yes. Sample answer: Let
f(x)=x+1 and g(x)=x1f\left(x\right)=x+1\text{ and }g\left(x\right)=x - 1
. Then
f(g(x))=f(x1)=(x1)+1=xf\left(g\left(x\right)\right)=f\left(x - 1\right)=\left(x - 1\right)+1=x
and
g(f(x))=g(x+1)=(x+1)1=xg\left(f\left(x\right)\right)=g\left(x+1\right)=\left(x+1\right)-1=x
. So
fg=gff\circ g=g\circ f
.

5. 
(f+g)(x)=2x+6\left(f+g\right)\left(x\right)=2x+6
, domain:
(,)\left(-\infty ,\infty \right)


(fg)(x)=2x2+2x6\left(f-g\right)\left(x\right)=2{x}^{2}+2x - 6
, domain:
(,)\left(-\infty ,\infty \right)


(fg)(x)=x42x3+6x2+12x\left(fg\right)\left(x\right)=-{x}^{4}-2{x}^{3}+6{x}^{2}+12x
, domain:
(,)\left(-\infty ,\infty \right)


(fg)(x)=x2+2x6x2\left(\frac{f}{g}\right)\left(x\right)=\frac{{x}^{2}+2x}{6-{x}^{2}}
, domain:
(,6)(6,6)(6,)\left(-\infty ,-\sqrt{6}\right)\cup \left(-\sqrt{6},\sqrt{6}\right)\cup \left(\sqrt{6},\infty \right)


7. 
(f+g)(x)=4x3+8x2+12x\left(f+g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}+1}{2x}
, domain:
(,0)(0,)\left(-\infty ,0\right)\cup \left(0,\infty \right)


(fg)(x)=4x3+8x212x\left(f-g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}-1}{2x}
, domain:
(,0)(0,)\left(-\infty ,0\right)\cup \left(0,\infty \right)


(fg)(x)=x+2\left(fg\right)\left(x\right)=x+2
, domain:
(,0)(0,)\left(-\infty ,0\right)\cup \left(0,\infty \right)


(fg)(x)=4x3+8x2\left(\frac{f}{g}\right)\left(x\right)=4{x}^{3}+8{x}^{2}
, domain:
(,0)(0,)\left(-\infty ,0\right)\cup \left(0,\infty \right)


9. 
(f+g)(x)=3x2+x5\left(f+g\right)\left(x\right)=3{x}^{2}+\sqrt{x - 5}
, domain:
[5,)\left[5,\infty \right)


(fg)(x)=3x2x5\left(f-g\right)\left(x\right)=3{x}^{2}-\sqrt{x - 5}
, domain:
[5,)\left[5,\infty \right)


(fg)(x)=3x2x5\left(fg\right)\left(x\right)=3{x}^{2}\sqrt{x - 5}
, domain:
[5,)\left[5,\infty \right)


(fg)(x)=3x2x5\left(\frac{f}{g}\right)\left(x\right)=\frac{3{x}^{2}}{\sqrt{x - 5}}
, domain:
(5,)\left(5,\infty \right)


11. a. 3; b.
f(g(x))=2(3x5)2+1f\left(g\left(x\right)\right)=2{\left(3x - 5\right)}^{2}+1
; c.
f(g(x))=6x22f\left(g\left(x\right)\right)=6{x}^{2}-2
; d.
(gg)(x)=3(3x5)5=9x20\left(g\circ g\right)\left(x\right)=3\left(3x - 5\right)-5=9x - 20
; e.
(ff)(2)=163\left(f\circ f\right)\left(-2\right)=163


13. 
f(g(x))=x2+3+2,g(f(x))=x+4x+7f\left(g\left(x\right)\right)=\sqrt{{x}^{2}+3}+2,g\left(f\left(x\right)\right)=x+4\sqrt{x}+7


15. 
f(g(x))=x+1x33=x+13x,g(f(x))=x3+1xf\left(g\left(x\right)\right)=\sqrt[3]{\frac{x+1}{{x}^{3}}}=\frac{\sqrt[3]{x+1}}{x},g\left(f\left(x\right)\right)=\frac{\sqrt[3]{x}+1}{x}


17. 
(fg)(x)=12x+44=x2, (gf)(x)=2x4\left(f\circ g\right)\left(x\right)=\frac{1}{\frac{2}{x}+4 - 4}=\frac{x}{2},\text{ }\left(g\circ f\right)\left(x\right)=2x - 4


19. 
f(g(h(x)))=(1x+3)2+1f\left(g\left(h\left(x\right)\right)\right)={\left(\frac{1}{x+3}\right)}^{2}+1


21. a.
(gf)(x)=324x\left(g\circ f\right)\left(x\right)=-\frac{3}{\sqrt{2 - 4x}}
; b.
(,12)\left(-\infty ,\frac{1}{2}\right)


23. a.
(0,2)(2,)\left(0,2\right)\cup \left(2,\infty \right)
; b.
(,2)(2,)\left(-\infty ,-2\right)\cup \left(2,\infty \right)
; c.
(0,)\left(0,\infty \right)


25. 
(1,)\left(1,\infty \right)


27. sample:
{f(x)=x3g(x)=x5\begin{cases}f\left(x\right)={x}^{3}\\ g\left(x\right)=x - 5\end{cases}


29. sample:
{f(x)=4xg(x)=(x+2)2\begin{cases}f\left(x\right)=\frac{4}{x}\qquad \\ g\left(x\right)={\left(x+2\right)}^{2}\qquad \end{cases}


31. sample:
{f(x)=x3g(x)=12x3\begin{cases}f\left(x\right)=\sqrt[3]{x}\\ g\left(x\right)=\frac{1}{2x - 3}\end{cases}


33. sample:
{f(x)=x4g(x)=3x2x+5\begin{cases}f\left(x\right)=\sqrt[4]{x}\\ g\left(x\right)=\frac{3x - 2}{x+5}\end{cases}


35. sample: 
f(x)=xf\left(x\right)=\sqrt{x}


g(x)=2x+6g\left(x\right)=2x+6


37.sample: 
f(x)=x3f\left(x\right)=\sqrt[3]{x}


g(x)=(x1)g\left(x\right)=\left(x - 1\right)


39. sample:
f(x)=x3f\left(x\right)={x}^{3}


g(x)=1x2g\left(x\right)=\frac{1}{x - 2}


41. sample:
f(x)=xf\left(x\right)=\sqrt{x}


g(x)=2x13x+4g\left(x\right)=\frac{2x - 1}{3x+4}


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))=94f\left(g\left(0\right)\right)=27,g\left(f\left(0\right)\right)=-94


75. 
f(g(0))=15,g(f(0))=5f\left(g\left(0\right)\right)=\frac{1}{5},g\left(f\left(0\right)\right)=5


77. 
18x2+60x+5118{x}^{2}+60x+51


79. 
gg(x)=9x+20g\circ g\left(x\right)=9x+20


81. 2

83. 
(,)\left(-\infty ,\infty \right)


85. False

87. 
(fg)(6)=6\left(f\circ g\right)\left(6\right)=6
;
(gf)(6)=6\left(g\circ f\right)\left(6\right)=6


89. 
(fg)(11)=11,(gf)(11)=11\left(f\circ g\right)\left(11\right)=11,\left(g\circ f\right)\left(11\right)=11


91. c. Solve
A(m(t))=4A\left(m\left(t\right)\right)=4
.

93. 
A(t)=π(25t+2)2A\left(t\right)=\pi {\left(25\sqrt{t+2}\right)}^{2}
and
A(2)=π(254)2=2500πA\left(2\right)=\pi {\left(25\sqrt{4}\right)}^{2}=2500\pi
square inches

95. 
A(5)=π(2(5)+1)2=121πA\left(5\right)=\pi {\left(2\left(5\right)+1\right)}^{2}=121\pi
square units

97. a.
N(T(t))=23(5t+1.5)256(5t+1.5)+1N\left(T\left(t\right)\right)=23{\left(5t+1.5\right)}^{2}-56\left(5t+1.5\right)+1
;

b. 3.38 hours

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