# Solutions 4: Composition of Functions

## Solutions for Try Its

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

No, the functions are not the same.

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

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

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

5. A. 8; B. 20

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

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

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

$f=h\circ g$

## 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.

$f\left(x\right)=x+1\text{ and }g\left(x\right)=x - 1$
. Then
$f\left(g\left(x\right)\right)=f\left(x - 1\right)=\left(x - 1\right)+1=x$
and
$g\left(f\left(x\right)\right)=g\left(x+1\right)=\left(x+1\right)-1=x$
. So
$f\circ g=g\circ f$
.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

17.
$\left(f\circ g\right)\left(x\right)=\frac{1}{\frac{2}{x}+4 - 4}=\frac{x}{2},\text{ }\left(g\circ f\right)\left(x\right)=2x - 4$

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

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

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

25.
$\left(1,\infty \right)$

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

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

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

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

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

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

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

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

39. sample:
$f\left(x\right)={x}^{3}$

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

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

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

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

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

77.
$18{x}^{2}+60x+51$

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

81. 2

83.
$\left(-\infty ,\infty \right)$

85. False

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

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

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

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

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

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

b. 3.38 hours