Loading [MathJax]/jax/element/mml/optable/GreekAndCoptic.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

2.4e: Exercises - Piecewise Functions, Combinations, Composition

( \newcommand{\kernel}{\mathrm{null}\,}\)

A: Concepts

Exercise 2.4e.A 

1) How does one find the domain of the quotient of two functions, fg?

2) What is the composition of two functions, fg?

3) If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

4) How do you find the domain for the composition of two functions, fg?

5) How do you graph a piecewise function?

Answers 1-5:

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)=x1. Then f(g(x))=f(x1)=(x1)+1=x and g(f(x))=g(x+1)=(x+1)1=x. So fg=gf.

5. Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x-axis and y-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate endpoints of −∞ or ∞.Combine the graphs to find the graph of the piecewise function

Piecewise Functions

B. Evaluate Piecewise Functions

Exercise 2.4e.B 

Given function f, evaluate f(3), f(2), f(1), and f(0).

6. f(x)={x+1if x<22x3if x2 7. f(x)={1if x30if x>3 8. f(x)={2x2+3if x15x7if x>1

Given function f, evaluate f(1), f(0), f(2), and f(4).

9. f(x)={7x+3if x<07x+6if x0 10. f(x)={x22if x<24+|x5|if x2 11. f(x)={5xif x<03if 0x2x2if x>3

Write the domain for each piecewise function in interval notation.

12. f(x)={x+1if x<22x3if x2 13. f(x)={x22if x<1x2+2if x>1 14. f(x)={x23if x<03x2if x2

15. Find f(5),f(0), and f(3) given f(x)={x2 if x0x+2 if x>0

16. Find f(3),f(0), and f(2) given f(x)={x3 if x<02x1 if x0

17. Find g(1),g(1), and g(4) given g(x)={5x2 if x<1x if x1

18. Find g(3),g(2), and g(1) given g(x)={x3 if x2|x| if x>2

19. Find h(2),h(0), and h(4) given  h(x)={5 if x<02x3 if 0x<2x2 if x2

20. Find h(5),h(4), and h(25) given h(x)={3x if x0x3 if 0<x4x if x>4

21. Find f(2),f(0), and f(3) given f(x)=[[x0.5]]

22. Find f(1.2),f(0.4), and f(2.6) given f(x)=[[2x]]+1

Answers to Odd Exercises:

7. f(3)=1; f(2)=0; f(1)=0; f(0)=0

9. f(1)=4; f(0)=6; f(2)=20; f(4)=34

11. f(1)=5; f(0)=3; f(2)=3; f(4)=16

13. domain: (,1)(1,)

15. f(5)=25,f(0)=0, and f(3)=5

17. g(1)=7,g(1)=1, and g(4)=2

19. h(2)=5,h(0)=3, and h(4)=16

21. f(2)=3,f(0)=1, and f(3)=2

C: Graph Piecewise Functions

Exercise 2.4e.C 

 Graph two-part piecewise functions.

  1. h(x)={x2+2 if x<0x+2 if x0
  2. h(x)={x23 if x<0x3 if x0
  3. h(x)={x31 if x<0|x3|4 if x0
  4. h(x)={x3 if x<0(x1)21 if x0
  5. h(x)={x21 if x<02 if x0
  6. h(x)={x+2 if x<0(x2)2 if x0
  1. g(x)={2 if x<0x if x0
  2. g(x)={x2 if x<03 if x0
  3. h(x)={x if x<0x if x0
  4. h(x)={|x| if x<0x3 if x0
  5. f(x)={|x| if x<24 if x2
  1. f(x)={x if x<1x if x1
  2. g(x)={x2 if x1x if x>1
  3. g(x)={3 if x1x3 if x>1
  4. h(x)={0 if x01x if x>0
  5. h(x)={1x if x<0x2 if x0
Answers to Odd Exercises:

23.

1790a36f5e4c391f1d37b3abdabb2349.png

25.

f2e8945e9fea8dc040b6d5a1180fd1d0.png

27.

ab01cd028abe7241da8e857be88bdb8a.pngFigure 2.4.27

29.

Figure 2.4.29
 

31.

Figure 2.4.31

.

33.

Figure 2.4.33

.

35.

Figure 2.4.35

.

37.

2.4e #37.png
Figure 2.4.37

.

 Graph 3 or more part piecewise functions.

  1. h(x)={(x+10)24 if x<8x+4 if 8x<4x+4 if x4
  2. f(x)={x+10 if x10|x5|15 if 10<x2010 if x>20
  3. f(x)={x2 if x<0x if 0x<22 if x2
  4. f(x)={x if x<1x3 if 1x<13 if x1
  5. g(x)={5 if x<2x2 if 2x<2x if x2
  1. g(x)={x if x<3|x| if 3x<1x if x1
  2. h(x)={1x if x<0x2 if 0x<24 if x2
  3. h(x)={0 if x<0x3 if 0<x28 if x>2
  4. f(x)=[[x+0.5]]
  5. f(x)=[[x]]]+1
  6. f(x)=[[0.5x]]
  7. f(x)=2[[x]]
Answers to Odd Exercises:
 
39.
00235242b3f9ae8ded77603b43125c75.png

Figure 2.4.39

41.

Figure 2.4.41

x

43.
Figure 2.4.43

45.

Figure 2.4.45

x

47.
Figure 2.4.47

49.

Figure 2.4.49

x

D: Graph Piecewise Functions and find their domain

Exercise 2.4e.D 

For each of the following, (a) graph the piecewise function, and (b) state its domain in interval notation.

51. f(x)={2x1if x<11+xif x1

52. f(x)={x+1if x<22x3if x2

53. f(x)={3if x<0xif x0

54. f(x)={x+1if x<0x1if x>0

55. f(x)={x2if x<0x+2if x0

56. f(x)={x2if x<01xif x>0

57. f(x)={|x|if x<21if x2

58. f(x)={x+1if x<1x3if x1

Answers to Odd Exercises:
51.

domain: (,)

Graph of f(x).

53.

domain: (,)

Graph of f(x).

55.

domain: (,)

Graph of f(x).

57.

domain: (,)

Graph of f(x).

E: Graph Piecewise Functions and evaluate them

Exercise 2.4e.E

 For each of the piecewise-defined functions, (a) sketch the graph, and (b) evaluate at the given values of the independent variable.

59. f(x)={x23,x04x3,x>0Find f(4),f(0),f(2)

60. f(x)={4x+3,x0x+1,x>0Find f(3),f(0),f(2)

61. g(x)={3x2,x24,x=2Find g(0),g(4),g(2)

62. h(x)=\begin{cases}x+1, &x≤5\\4, &x>5\end{cases} \quad \text{Find } h(0),  \; h(π), \; h(5)

Answers to Odd Exercises:
59.  f(−4)=13,\; f(0)=-3,\; f(2)=5
\qquadAn image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a function that has two pieces. The first piece is a decreasing curve that ends at the point (0, -3). The second piece is an increasing line that begins at the point (0, -3). The function has a x intercepts at the approximate point (1.7, 0) and the point (0.75, 0) and a y intercept at (0, -3).
61. a. g(0)=\frac{−3}{2},\; g(−4)=\frac{−1}{2},\; g(2)=4 
\qquadAn image of a graph. The x axis runs from -10 to 10 and the y axis runs from -10 to 10. The graph is of a function that begins slightly below the x axis and begins to decrease. As the function approaches the unplotted vertical line of “x = 2”, it decreases at a faster rate but never reaches the line “x = 2”. On the right side of the unplotted line “x = 2”, the function starts at the top of graph and begins decreasing and approaches the unplotted horizontal line “y = 0”, but never reaches “y = 0”. There function also includes a plotted point at (2, 4). There is a y intercept at (0, -1.5) and no x intercept.

F: Construct the equation for a piecewise function given a graph 

Exercise \PageIndex{F}  

\bigstar (a) Evaluate piecewise function values from a graph. (b) Construct a piecewise function corresponding to the graph.

63. Find f(-4), f(-2), and f(0).

Figure 2.4e.63

64. Find f(−3), f(0), and f(1).

Figure 2.4e.64

65. Find f(0), f(2), and f(4).

Figure 2.4e.65

66. Find f(−5), f(−2), and f(2).

Figure 2.4e.66

67. Find f(−3), f(−2), and f(2).

Figure 2.4e.67

68. Find f(−3), f(0), and f(4).

Figure 2.4e.68

69. Find f(−2), f(0), and f(2).

Figure 2.4e.69

70. Find f(−3), f(1), and f(2).

Figure 2.4e.70
 
Answers to Odd Exercises:

63. f(−4) = 1, f(−2) = 1, and f(0) = 0 \qquad  f(x)=\begin{cases}1, & x≤-2\\ x, & x>-2\end{cases} )

65. f(0) = 0, f(2) = 8, and f(4) = 0 \qquad \qquad  f(x)=\begin{cases}-2, & x≤0\\ \frac{1}{x}, & x>0\end{cases}

67. f(−3) = 5, f(−2) = 4, and f(2) = 2 \qquad  f(x)=\begin{cases}5, & x< -2\\x^2, & -2 \le x < 2\\ x, & x \ge 2\end{cases} 

69. f(−2) = −1, f(0) = 0, and f(2) = 1 \qquad  f(x)=\begin{cases}-1, & x<0\\ 0, & x=0\\ 1, & x>0\end{cases} 

G: Simplify Combination Functions and Find their Domains

Exercise \PageIndex{G} 

  \bigstar  For each pair of functions f and g given below, find and simplify the combination functions f+g, f−g, fg, and \dfrac{f}{g}. State the domain of each combination functions in interval notation.

71. f(x)=x^2+2x,  g(x)=6−x^2 \\[4pt] .

72. f(x)=−3x^2+x,  g(x)=5.

73. f(x)=2x^2+4x,  g(x)=\dfrac{1}{2x}.

74. f(x)=\dfrac{1}{x−4},  g(x)=\dfrac{1}{6−x}.

75. f(x)=3x^2,  g(x)=\sqrt{x−5}.

76. f(x)=\sqrt{x},  g(x)=|x−3|

Answers to Odd Exercises:
  1. (f+g)(x)=2x+6, domain: (−\infty,\infty)
    (f−g)(x)=2x^2+2x−6, domain: (−\infty,\infty) 
    (fg)(x)=−x^4−2x^3+6x^2+12x, domain: (−\infty,\infty) 
    \left(\dfrac{f}{g}\right)(x)=\dfrac{x^2+2x}{6−x^2}, domain:  (−\infty,−\sqrt{6})\cup(\sqrt{6},\sqrt{6})\cup(\sqrt{6},\infty)

 

  1. (f+g)(x)=\dfrac{4x^3+8x^2+1}{2x}, domain: (−\infty,0)\cup(0,\infty) 
    (f−g)(x)=\dfrac{4x^3+8x^2−1}{2x}, domain: (−\infty,0)\cup(0,\infty) 
    (fg)(x)=x+2, domain: (−\infty,0)\cup(0,\infty) 
    \left(\dfrac{f}{g}\right) (x)=4x^3+8x^2, domain: (−\infty,0)\cup(0,\infty)

 

  1. (f+g)(x)=3x^2+\sqrt{x−5}, domain: \left[5,\infty\right) 
    (f−g)(x)=3x^2−\sqrt{x−5}, domain: \left[5,\infty\right) 
    (fg)(x)=3x^2\sqrt{x−5}, domain: \left[5,\infty\right) 
    \left(\dfrac{f}{g}\right)(x)=\dfrac{3x^2}{\sqrt{x−5}}, domain: (5,\infty)

 

 

Composition

H: Evaluate a Composition from Tables

Exercise \PageIndex{H} 

\bigstar  Use the function values for f and g shown in the table  below to evaluate each expression.

x 0 1 2 3 4 5 6 7 8 9
f(x) 7 6 5 8 4 0 2 1 9 3
g(x) 9 5 6 2 1 8 7 3 4 0

78. f(g(8)) \; 79. f(g(5)) \; 80. g(f(5)) \; 81. g(f(3)) \; 82. f(f(4)) \; 83. f(f(1)) \; 84. g(g(2)) \; 85. g(g(6))

\bigstar  Use the function values for f and g shown in the table  below to evaluate each expression.

x -3 -2 -1 0 1 2 3
f(x) 11 9 7 5 3 1 -1
g(x) -8 -3 0 1 0 -3 -8

 86. (f{\circ}g)(1) \quad 87. (f{\circ}g)(2) \quad 88. (g{\circ}f)(2) \quad 89. (g{\circ}f)(3) \quad 90. (g{\circ}g)(1) \quad 91. (f{\circ}f)(3)

Answers to Odd Exercises:
79. 9 81. 4 83. 2 85. 3 87. 11 89. 0 91. 7

I: Evaluate a Composition from Graphs

Exercise \PageIndex{I} 

\bigstar  Use graphs to evaluate the following compositions.

92.  (f \circ g ) (3) \\[5pt]

92.1  (f \circ g ) (6) \\[5pt]

93.  (f \circ g ) (1) \\[5pt]

94.  (g \circ f ) (1) \\[5pt]

95.  (g \circ f ) (0)

96.  (f \circ f ) (5) \\[5pt]

97.  (f \circ f ) (4) \\[5pt]

98.  (g \circ g ) (2) \\[5pt]

99.  (g \circ g ) (0)

f
Graph of a function.
g
Graph of a function.

  \bigstar  Use graphs to evaluate the following compositions.

100. g(f(1)) \\[5pt]

101. g(f(2)) \\[5pt]

102. f(g(4)) \\[5pt]

103. f(g(1)) \\[5pt]

104. f(h(2)) \\[5pt]

105. h(f(2)) \\[5pt]

106. f(g(h(4))) \\[5pt]

107. f(g(f(−2)))

Graph of a parabola      Graph of a square root function.

Graph of an absolute value function.

Answers to Odd Exercises:
93. 2 95. 5 97. 4 99. 0 101. 2 103. 1 105. 4 107. 4

J: Evaluate a Composition from Formulas

Exercise \PageIndex{J} 

\bigstar  Use the given pair of functions to find the following values if they exist.

 a.  (g\circ f)(0)  b.  (f\circ g)(-1) c.  (f \circ f)(2)  d.  (g\circ f)(-3) e.  (f\circ g)\left(\frac{1}{2}\right) f.  (f \circ f)(-2)
  1.  f(x) = x^2 , g(x) = 2x+1 \\[5pt]  
  2.  f(x) = 4-x , g(x) = 1-x^2 \\[5pt]
  3.  f(x) = 4-3x , g(x) = |x| \\[5pt]
  4.  f(x) = |x-1| , g(x) = x^2-5
  1. f(x) = 4x+5 , g(x) = \sqrt{x} \\[5pt]
  2.  f(x) = \sqrt{3-x} , g(x) = x^2+1 \\[5pt]
  3.  f(x) = 6-x-x^2 , g(x) = x\sqrt{x+10} \\[5pt]
  4.  f(x) = \sqrt[3]{x+1} , g(x) = 4x^2-x \\[5pt]
  1. f(x) = \dfrac{3}{1-x} , g(x) = \dfrac{4x}{x^2+1}
  2.  f(x) = \dfrac{x}{x+5} , g(x) = \dfrac{2}{7-x^2}
  3.  f(x) = \dfrac{2x}{5-x^2} , g(x) = \sqrt{4x+1}
  4.  f(x) =\sqrt{2x+5} , g(x) = \dfrac{10x}{x^2+1}
Answers to Odd Exercises

 111. f(x) = x^2 , g(x) = 2x+1 :

  1.  (g\circ f)(0) = 1
  2.  (f\circ g)(-1) = 1
  3.  (f \circ f)(2) = 16
  4.  (g\circ f)(-3) = 19
  5.  (f\circ g)\left(\frac{1}{2}\right) = 4
  6.  (f \circ f)(-2) = 16

 

113.  f(x) = 4-3x , g(x) = |x| :

  1.  (g\circ f)(0) = 4
  2.  (f\circ g)(-1) = 1
  3.  (f \circ f)(2) = 10
  4. (g\circ f)(-3) = 13
  5.  (f\circ g)\left(\frac{1}{2}\right) = \frac{5}{2}
  6.  (f \circ f)(-2) = -26

 115. f(x) = 4x+5 , g(x) = \sqrt{x} :

  1.  (g\circ f)(0) = \sqrt{5}
  2.  (f\circ g)(-1) is not real
  3.  (f \circ f)(2) = 57
  4.  (g\circ f)(-3) is not real
  5.  (f\circ g)\left(\frac{1}{2}\right) = 5+2\sqrt{2}
  6.  (f \circ f)(-2) = -7

 

117. f(x)=6-x-x^2 ,
\quad g(x)=x\sqrt{x+10}

  1.  (g\circ f)(0) = 24
  2.  (f\circ g)(-1) = 0
  3.  (f \circ f)(2) = 6
  4.  (g\circ f)(-3) = 0
  5.  (f\circ g)\left(\frac{1}{2}\right) = \frac{27-2\sqrt{42}}{8}
  6.  (f \circ f)(-2) = -14

119. f(x) = \frac{3}{1-x} , g(x) = \frac{4x}{x^2+1} :

  1.  (g\circ f)(0) = \frac{6}{5}
  2.  (f\circ g)(-1) = 1
  3.  (f \circ f)(2) = \frac{3}{4}
  4.  (g\circ f)(-3) = \frac{48}{25}
  5.  (f\circ g)\left(\frac{1}{2}\right) = -5
  6.  (f \circ f)(-2) is undefined

 

121. f(x) = \frac{2x}{5-x^2} , g(x) = \sqrt{4x+1} :

  1.  (g\circ f)(0) = 1
  2.  (f\circ g)(-1) is not real
  3.  (f \circ f)(2) = -\frac{8}{11}
  4.  (g\circ f)(-3) = \sqrt{7}
  5.  (f\circ g)\left(\frac{1}{2}\right) = \sqrt{3}
  6.  (f \circ f)(-2) = \frac{8}{11}

K: Simplify a Composition and Find its Domain

Exercise \PageIndex{K}: Find and simplify the Equation for a Composition

\bigstar Find and simplify (a) (f \circ g)(x), and (b) (g \circ f)(x). State the domain for (c) (f \circ g)(x) and for (d) (g \circ f)(x).

  1. f(x)=x^5,  g(x)=x+1 \\[5pt]
  2. f(x)=|x|, g(x)=5x+1 \\[5pt]
  3. f(x) = 2x+3 , g(x) = x^2-9 \\[5pt]
  4. f(x)=4x+8, g(x)=7−x^2 \\[5pt]
  5. f(x)=5x+7, g(x)=4−2x^2 \\[5pt]
  1. f(x)=2x^2+1, g(x)=3x+5 \\[5pt]
  2. f(x)=2x^2+1g(x)=3x−5 \\[5pt]
  3. f(x) = x^2 -x+1 , g(x) = 3x-5 \\[5pt]
  4. f(x) = x^2-4 , g(x) = |x| \\[5pt]
Answers to Odd Exercises

127.  a. (f \circ g )(x) = (x+1)^5, domain:  (−\infty,\infty) \qquad b.  (g \circ f ) (x)= x^5+1 . domain:  (−\infty,\infty)
129.  a. (f \circ g )(x)= 2x^2-15 , domain: (-\infty, \infty)   \qquad b.  (g \circ f ) (x)= 4x^2+12x , domain: (-\infty, \infty)
131.  a. (f \circ g )(x)= 27-10x^2, domain:  (−\infty,\infty)   \;  b.  (g \circ f ) (x)=-50x^2-140x-94 . domain:  (−\infty,\infty)
133.  a. (f \circ g )(x)= 2(3x−5)^2+1, domain:  (−\infty,\infty)   \qquad b.  (g \circ f ) (x)= 6x^2−2. domain:  (−\infty,\infty)
135.  a. (f \circ g )(x)= x^2-4 , domain: (-\infty, \infty)   \qquad b.  (g \circ f ) (x)= |x^2-4| , domain: (-\infty, \infty)

\bigstar Find and simplify (a) (f \circ g)(x), and (b) (g \circ f)(x). State the domain for (c) (f \circ g)(x) and for (d) (g \circ f)(x).

  1. f(x) = 3x-5 , g(x) = \sqrt{x} \\[5pt]
  2. f(x)=\sqrt{x}+2, g(x)=x^2+3 \\[5pt]
  3. f(x) = |x+1| , g(x) = \sqrt{x} \\[5pt]
  4. f(x) = |x| , g(x) = \sqrt{4-x} \\[5pt]
  5. f(x)=x^2+2,  g(x)=\sqrt{x−2} \\[5pt]
  6. f(x)=x^2+1, g(x)=\sqrt{x+2} \\[5pt]
  1. f(x) = x^2-x-1 , g(x) = \sqrt{x-5} \\[5pt]
  2. f(x) = 3-x^2 , g(x) = \sqrt{x+1} \\[5pt]
  3. f(x)=\dfrac{1}{\sqrt{x}},  g(x)=x^2−4 \\[5pt]
  4. f(x)=\dfrac{1}{\sqrt{x}},  g(x)=x^2−9 \\[5pt]
  5. f(x)=\sqrt{x+4}, g(x)=12−x^3 \\[5pt]
  6. f(x)=x^3+1 and g(x)=\sqrt[3]{x−1}
Answers to Odd Exercises

137.   a. (f \circ g )(x)= 3 \sqrt{x} -5, domain: [0, \infty)   \qquad b.  (g \circ f ) (x)= \sqrt{3x-5} . domain: [ \frac{5}{3}, \infty )
139.   a. (f \circ g )(x)= \sqrt{x}+1 , domain: [0,\infty) \qquad b.  (g \circ f ) (x)= \sqrt{|x+1|} , domain: (-\infty, \infty)
141.  a. (f \circ g )(x)= x, domain: [2, \infty) \qquad b.  (g \circ f ) (x)= |x| . domain:  (−\infty,\infty)
143.   a. (f \circ g )(x)= x-6- \sqrt{x-5}, d:  [5,\infty) \qquad b.  (g \circ f ) (x)= \sqrt{x^2-x-6} . d:   (−\infty,-2]\cup[3,\infty)
145.  a. (f \circ g )(x)= \frac{1}{\sqrt{x^2-4}}, domain:  (−\infty,−2)\cup(2,\infty) \qquad b.  (g \circ f ) (x)= \frac{1}{x}-4 . domain:  (0,\infty)
147.  a. (f \circ g )(x)= \sqrt{16-x^3}, d: ( -\infty, 2\sqrt[3]{2} ]  \qquad b.  (g \circ f ) (x)= 12-(x+4) \sqrt{x+4}. d: [-4, \infty)

\bigstar Find and simplify (a) (f \circ g)(x), and (b) (g \circ f)(x). State the domain for (c) (f \circ g)(x) and for (d) (g \circ f)(x).

  1. f(x)=\dfrac{1}{x},  g(x)=x−3 \\[5pt]
  2. f(x)=\frac{1}{x+2}, g(x)=4x+3 \\[5pt]
  3. f(x) = 3x-1 , g(x) = \dfrac{1}{x+3} \\[5pt]
  4. f(x)=\dfrac{1}{x−6}, g(x)=\dfrac{7}{x}+6 \\[5pt]
  5. f(x)=\dfrac{1}{x−4}, g(x)=\dfrac{2}{x}+4 \\[5pt]
  6. f(x) = \dfrac{3x}{x-1} , g(x) =\dfrac{x}{x-3} \\[5pt]
  1. f(x) = \dfrac{x}{2x+1} , g(x) = \dfrac{2x+1}{x} \\[5pt]
  2. f(x)=\dfrac{1−x}{x},  g(x)=\dfrac{1}{1+x^2} \\[5pt]
  3. f(x)=\dfrac{1}{x},  g(x)=\sqrt{x−1} \\[5pt]
  4. f(x)=\sqrt{2−4x},  g(x)=−\dfrac{3}{x} \\[5pt]
  5. f(x)=\sqrt[3]{x}, g(x)=\dfrac{x+1}{x^3} \\[5pt]
  6. f(x) = \dfrac{2x}{x^2-4} , g(x) =\sqrt{1-x} \\[5pt]
Answers to Odd Exercises:

149.  a. (f \circ g )(x)= \frac{1}{x-3}, domain:  (−\infty,3)\cup(3,\infty) \qquad b.  (g \circ f ) (x)=\frac{1}{x}-3 . domain:  (−\infty,0)\cup(0,\infty)
151.   a. (f \circ g )(x)= -\frac{x}{x+3} , d: \left(-\infty, -3\right) \cup \left(-3, \infty\right) \qquad b.  (g \circ f ) (x)= \frac{1}{3x+2} , d: \left(-\infty, -\frac{2}{3}\right) \cup \left(-\frac{2}{3}, \infty\right)
153.  a. (f \circ g )(x)= \dfrac{x}{2} , domain:  (−\infty,0)\cup(0,\infty) \qquad b.  (g \circ f ) (x)=2x-4 . domain:  (−\infty,4)\cup(4,\infty)
155.   a. (f \circ g )(x)= \frac{2x+1}{5x+2} , d: \left(-\infty, -\frac{2}{5}\right) \cup \left(-\frac{2}{5}, 0\right) \cup (0,\infty)
\qquad \; b.  (g \circ f ) (x)=\frac{4x+1}{x} , domain: \left(-\infty, -\frac{1}{2}\right) \cup \left(-\frac{1}{2}, 0), \cup (0, \infty\right)
157.  a. (f \circ g )(x)= \frac{1}{\sqrt{x-1}}, domain:  (1,\infty) \qquad b.  (g \circ f ) (x)= \sqrt{\frac{1}{x}-1}. domain: (0, 1]
159.  a. (f \circ g )(x)= \dfrac{\sqrt[3]{x+1}}{x} , d:  (−\infty,0)\cup(0,\infty) \qquad b.  (g \circ f ) (x)= \frac{\sqrt[3]{x}+1}{x} . d:  (−\infty,0)\cup(0,\infty)

\bigstar (a) Find and simplify (f \circ f)(x) \ and (b) state the domain of the composition.

  1.  f(x) = 2x+3  \\[5pt]  
  2.  f(x) = x^2 -x+1   \\[5pt]
  3.  f(x) = x^2-4   \\[5pt]
  4.  f(x) = 3x-5 
  1.  f(x) = |x+1|   \\[5pt]
  2.  f(x) = 3-x^2   \\[5pt]
  3.  f(x) = |x|   \\[5pt]
  4.  f(x) = x^2-x-1   
  1.  f(x) = 3x-1
  2.  f(x) = \dfrac{3x}{x-1}  
  3.  f(x) = \dfrac{x}{2x+1}   
  4.  f(x) = \dfrac{2x}{x^2-4}  

\bigstar Given  f(x) = -2x , g(x) = \sqrt{x} and h(x) = |x| , find and simplify expressions for the following functions and state the domain of each using interval notation.

  1.  (h\circ g \circ f)(x)
  2.  (h\circ f \circ g)(x)
  1.  (g\circ f \circ h)(x)
  2.  (g\circ h \circ f)(x)
  1.  (f\circ h \circ g)(x)
  2.  (f\circ g \circ h)(x)  
Answers to Odd Exercises:

163.  (f \circ f)(x) = 4x+9 , domain: (-\infty, \infty)
165.  (f \circ f)(x) =x^4-8x^2+12 , domain: (-\infty, \infty)
167. (f \circ f)(x) = ||x+1|+1| = |x+1|+1 , domain: (-\infty, \infty)
169. (f \circ f)(x) = | |x| | = |x| , domain: (-\infty, \infty)
171.  (f \circ f)(x) = 9x-4 , domain: (-\infty, \infty)
173.  (f \circ f)(x) = \frac{x}{4x+1} , d: \left(-\infty, -\frac{1}{2}\right) \cup \left(-\frac{1}{2}, -\frac{1}{4} \right) \cup \left(-\frac{1}{4},\infty\right)
175. (h\circ g \circ f)(x)= |\sqrt{-2x}|= \sqrt{-2x} , domain: (-\infty, 0]
177. (g\circ f \circ h)(x) = \sqrt{-2|x|} , domain: \{0\}
179. (f\circ h \circ g)(x) = -2|\sqrt{x}| = -2\sqrt{x} , domain: [0,\infty)

L: Decomposition

Exercise \PageIndex{L} 

\bigstar  Find functions f(x) and g(x) so the given function can be expressed as f(g(x)).

  1. h(x)=\sqrt { \dfrac{2x−1}{3x+4}}
  2. h(x)=(x+2)^2
  3. h(x)=(x−5)^3
  4. h(x)=\dfrac{3}{x−5}
  5. h(x)=\dfrac{4}{(x+2)^2}
  6. h(x)=4+\sqrt[3]{x}
  7. h(x)=\sqrt[3]{\dfrac{1}{2x−3}}
  8. h(x)=\dfrac{1}{(3x^2−4)^{−3}}
  1. h(x)=\sqrt[4]{\dfrac{3x−2}{x+5}}
  2. h(x)=\left(\dfrac{8+x^3}{8−x^3}\right)^4
  3. h(x)=\sqrt{2x+6}
  4. h(x)=(5x−1)^3
  5. h(x)=\sqrt[3]{x−1}
  6. h(x)=|x^2+7|
  7. h(x)=\dfrac{1}{(x−2)^3}
  8. h(x)=\left(\dfrac{1}{2x−3}\right)^2
  9. p(x) = (2x+3)^3  
  1. P(x) = \left(x^2-x+1\right)^5
  2. h(x) = \sqrt{2x-1}
  3. H(x) = |7-3x|
  4.  r(x) = \dfrac{2}{5x+1}
  5.  R(x) = \dfrac{7}{x^2-1}
  6.  q(x) = \dfrac{|x|+1}{|x|-1}
  7.  Q(x) = \dfrac{2x^3+1}{x^3-1}
  8.  v(x) = \dfrac{2x+1}{3-4x}
  9.  w(x) = \dfrac{x^2}{x^4+1}
Answers to Odd Exercises:

185. sample: f(x)=\sqrt{x}, \quad g(x)=\frac{2x−1}{3x+4}
187. sample: f(x)=x^3, \quad g(x)=x−5
189: sample: f(x)=\frac{4}{x}, \quad g(x)=(x+2)^2
191. sample: f(x)=\sqrt[3]{x}, \quad g(x)=\frac{1}{2x−3}
193. sample: f(x)=\sqrt[4]{x}, \quad g(x)=\frac{3x−2}{x+5}
195. sample: f(x)=\sqrt{x}, \quad g(x)=2x+6
197. sample: f(x)=\sqrt[3]{x}, \quad g(x)=(x−1)
199. sample: f(x)=x^3, \quad g(x)=\frac{1}{x−2}
201. Let g(x) = 2x+3 and f(x) = x^3 , then p(x) = (f\circ g)(x) .
203. Let g(x) = 2x-1 and f(x) = \sqrt{x} , then h(x) = (f\circ g)(x) .
205. Let g(x) = 5x+1 and f(x) = \frac{2}{x} , then r(x) =(f\circ g)(x) .
207. Let g(x) = |x| and f(x) = \frac{x+1}{x-1} , then q(x) =(f\circ g)(x) .
209. Let g(x) =2x and f(x) = \frac{x+1}{3-2x} , then v(x) =(f\circ g)(x) .


2.4e: Exercises - Piecewise Functions, Combinations, Composition is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

  • Was this article helpful?

Support Center

How can we help?