2.3.2: Homework

Algebraic

For the following exercises, use the pair of functions \(\ f\) and \(\ g\) to find the following values if they exist. Simplify your answers.

• \(\ (f + g)(2)\)
• \(\ (f − g)(−1) \)
• \(\ (g − f)(1)\)
• \(\ (f g)\left(\frac{1}{2}\right)\)
• \(\ \left(\frac{f}{g}\right)(0)\)
• \(\ \left(\frac{g}{f}\right)(-2)\)

15. \(\ f(x) = 3x + 1\) and \(\ g(x) = 4 − x\)
16. \(\ f(x)=x^{2}\) and \(\ g(x) = −2x + 1\)
17. \(\ f(x)=x^{2}-x\) and \(\ g(x)=12-x^{2}\)
18. \(\ f(x)=2 x^{3}\) and \(\ g(x)=-x^{2}-2 x-3\)
19. \(\ f(x)=\sqrt{x+3}\) and \(\ g(x) = 2x − 1\)
20. \(\ f(x)=\sqrt{4-x}\) and \(\ g(x)=\sqrt{x+2}\)
21. \(\ f(x)=2 x\) and \(\ g(x)=\frac{1}{2 x+1}\)
22. \(\ f(x)=x^{2}\) and \(\ g(x)=\frac{3}{2 x-3}\)
23. \(\ f(x)=x^{2}\) and \(\ g(x)=\frac{1}{x^{2}}\)
24. \(\ f(x)=x^{2}+1\) and \(\ g(x)=\frac{1}{x^{2}+1}\)

For the following exercises, use the pair of functions \(\ f\) and \(\ g\) to find the domain of the indicated function then find and simplify an expression for it.

• \(\ (f + g)(x)\)
• \(\ (f − g)(x)\)
• \(\ (fg)(x)\)
• \(\ \left(\frac{f}{g}\right)(x)\)

25. \(\ f(x) = 2x + 1\) and \(\ g(x) = x − 2\)
26. \(\ f(x) = 1 − 4x\) and \(\ g(x) = 2x − 1\)
27. \(\ f(x)=x^{2}\) and \(\ g(x) = 3x − 1\)
28. \(\ f(x)=x^{2}-x\) and \(\ g(x) = 7x\)
29. \(\ f(x)=x^{2}-4\) and \(\ g(x) = 3x + 6\)
30. \(\ f(x)=-x^{2}+x+6\) and \(\ g(x)=x^{2}-9\)
31. \(\ f(x)=\frac{x}{2}\) and \(\ g(x)=\frac{2}{x}\)
32. \(\ f(x) = x − 1\) and \(\ g(x)=\frac{1}{x-1}\)
33. \(\ f(x) = x\) and \(\ g(x)=\sqrt{x+1}\)
34. \(\ f(x)=\sqrt{x-5}\) and \(\ g(x)=f(x)=\sqrt{x-5}\)
35. Given \(f(x)= x^2 +2x\) and \(g(x)=6− x^2 \), find \(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g} \). Determine the domain for each function in interval notation.
36. Given \(f(x)=−3 x^2 +x\) and \(g(x)=5\), find \(f+g\), \( f−g \), \(fg\), and \(\frac{f}{g} \). Determine the domain for each function in interval notation.
37. Given \(f(x)=2 x^2 +4x\) and \(g(x)= \frac{1}{2x} \), find \(f+g\), \( f−g \), \(fg\), and \(\frac{f}{g} \). Determine the domain for each function in interval notation.
38. Given \(f(x)= \frac{1}{x−4}\) and \(g(x)= \frac{1}{6−x} \), find \(f+g\), \( f−g \), \(fg\), and \(\frac{f}{g} \). Determine the domain for each function in interval notation.
39. Given \(f(x)=3 x^2\) and \(g(x)=\sqrt{ x−5 } \), find \(f+g\), \( f−g \), \(fg\), and \(\frac{f}{g} \). Determine the domain for each function in interval notation.
40. Given \(f(x)=\sqrt{ x }\) and \(g(x)=|x−3|\), find \(\frac{g}{f} \). Determine the domain of the function in interval notation.
41. Given \(f(x)=2 x^2 +1\) and \(g(x)=3x−5\), find the following:
    1. \(f(g(2))\)
    2. \(f(g(x))\)
    3. \(g(f(x))\)
    4. \(( g \circ g )( x )\)
    5. \(( f \circ f )( −2 )\)

For the following exercises, find and simplify the difference quotient \(\ \frac{f(x+h)-f(x)}{h}\) for the given function.

42. \(\ f(x) = 2x − 5\)
43. \(\ f(x) = −3x + 5\)
44. \(\ f(x) = 6\)
45. \(\ f(x)=3 x^{2}-x\)
46. \(\ f(x)=-x^{2}+2 x-1\)
47. \(\ f(x)=4 x^{2}\)
48. \(\ f(x)=x-x^{2}\)
49. \(\ f(x)=x^{3}+1\)
50. \(\ f(x) = mx + b\) where \(\ m \neq 0\)
51. \(\ f(x)=a x^{2}+b x+c\) where \(\ a \neq 0\)
52. \(\ f(x)=\frac{2}{x}\)
53. \(\ f(x)=\frac{3}{1-x}\)
54. \(\ f(x)=\frac{1}{x^{2}}\)
55. \(\ f(x)=\frac{2}{x+5}\)
56. \(\ f(x)=\frac{1}{4 x-3}\)
57. \(\ f(x)=\frac{3 x}{x+1}\)
58. \(\ f(x)=\frac{x}{x-9}\)
59. \(\ f(x)=\frac{x^{2}}{2 x+1}\)
60. \(\ f(x)=\sqrt{x-9}\)
61. \(\ f(x)=\sqrt{2 x+1}\)
62. \(\ f(x)=\sqrt{-4 x+5}\)
63. \(\ f(x)=\sqrt{4-x}\)
64. \(\ f(x)=\sqrt{a x+b}\), where \(\ a \neq 0\).
65. \(\ f(x)=x \sqrt{x}\)
66. \(\ f(x)=\sqrt[3]{x}\).
    HINT: \(\ (a-b)\left(a^{2}+a b+b^{2}\right)=a^{3}-b^{3}\)

For the following exercises, use each pair of functions to find \(f( g( x ) )\) and \(g( f( x ) )\). Simplify your answers.

67. \(f(x)= x^2 +1\), \(g(x)=\sqrt{ x+2 }\)
68. \(f(x)=\sqrt{ x } +2\), \(g(x)= x^2 +3\)
69. \(f(x)=| x |\), \(g(x)=5x+1\)
70. \(f(x)= \sqrt[3]{x}\), \(g(x)= \frac{x+1}{x^3}\)
71. \(f(x)= \frac{1}{x−6}\), \(g(x)= \frac{7}{x} +6\)
72. \(f(x)= \frac{1}{x−4}\), \(g(x)= \frac{2}{x} +4\)

For the following exercises, use each set of functions to find \(f( g( h(x) ) )\). Simplify your answers.

73. \(f(x)= x^4 +6\), \(g(x)=x−6\), and \(h(x)=\sqrt{ x }\)
74. \(f(x)= x^2 +1\), \(g(x)= \frac{1}{x} \), and \(h(x)=x+3\)
75. Given \(f(x)= \frac{1}{x}\) and \(g(x)=x−3\), find the following:
    1. \((f \circ g)(x)\)
    2. the domain of \((f \circ g)(x)\) in interval notation
    3. \((g \circ f)(x)\)
    4. the domain of \((g \circ f)(x)\)
    5. \(\left( \frac{f}{g} \right)x\)
76. Given \(f(x)=\sqrt{ 2−4x }\) and \(g(x)=−\frac{3}{x} \), find the following:
    1. \((g \circ f)(x)\)
    2. the domain of \((g \circ f)(x)\) in interval notation
77. Given the functions \( f(x)= \frac{1−x}{x} \) and \(g(x)= \frac{1}{1+ x^2}\), find the following:
    1. \((g \circ f)(x)\)
    2. \((g \circ f)(\text{2})\)
78. Given functions \(p(x)= \frac{1}{\sqrt{ x }}\) and \(m(x)= x^2 −4\), state the domain of each of the following functions using interval notation:
    1. \(p(x) m(x)\)
    2. \(p(m(x))\)
    3. \(m(p(x))\)
79. Given functions \(q(x)= \frac{1}{\sqrt{ x }}\) and \(h(x)= x^2 −9\), state the domain of each of the following functions using interval notation.
    1. \( \frac{q(x)}{h(x)}\)
    2. \(q( h(x) )\)
    3. \(h( q(x) )\)
80. For \(f(x)= \frac{1}{x}\) and \(g(x)=\sqrt{ x−1 } \), write the domain of \((f \circ g)(x)\) in interval notation.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f( g(x) )\).

81. \(h(x)= (x+2)^2\)
82. \(h(x)= (x−5)^3\)
83. \(h(x)= \frac{3}{x−5}\)
84. \(h(x)= \frac{4}{(x+2)^2}\)
85. \(h(x)=4+ \sqrt[3]{x}\)
86. \(h(x)= \sqrt[3]{\frac{1}{2x−3}}\)
87. \(h(x)= \frac{1}{(3 x^2 −4)^{−3}}\)
88. \(h(x)= \sqrt[4]{\frac{3x−2}{x+5}}\)
89. \(h(x)= \left( \frac{8+ x^3}{8− x^3} \right)^4\)
90. \(h(x)=\sqrt{ 2x+6 }\)
91. \(h(x)= (5x−1)^3\)
92. \(h(x)= \sqrt[3]{x−1}\)
93. \(h(x)=| x^2 +7 |\)
94. \(h(x)= \frac{1}{(x−2)^3}\)
95. \(h(x)= \left( \frac{1}{2x−3} \right)^2\)
96. \(h(x)=\sqrt{ \frac{2x−1}{3x+4} }\)

Graphical

For the following exercises, use the graphs of \(f\), shown in Figure \( \PageIndex{ 1 } \), and \(g\), shown in Figure \( \PageIndex{ 2 } \), to evaluate the expressions.

97. \(f( g(3) )\)
98. \(f( g(1) )\)
99. \(g( f(1) )\)
100. \(g( f(0) )\)
101. \(f( f(5) )\)
102. \(f( f(4) )\)
103. \(g( g(2) )\)
104. \(g( g(0) )\)

For the following exercises, use graphs of \(f(x)\), shown in Figure \( \PageIndex{ 3 } \), \(g(x)\), shown in Figure \( \PageIndex{ 4 } \), and \(h(x)\), shown in Figure \( \PageIndex{ 5 } \), to evaluate the expressions.

105. \(g( f( 1 ) )\)
106. \(g( f( 2 ) )\)
107. \(f( g( 4 ) )\)
108. \(f( g( 1 ) )\)
109. \(f( h( 2 ) )\)
110. \(h( f( 2 ) )\)
111. \(f( g( h( 4 ) ) )\)
112. \(f( g( f( −2 ) ) )\)

Numeric

For the following exercises, use the function values for \(f\) and \(g\) shown in Table \( \PageIndex{ 1 } \) to evaluate each expression.

113. \(f( g( 8 ) )\)
114. \(f( g( 5 ) )\)
115. \(g( f( 5 ) )\)
116. \(g( f( 3 ) )\)
117. \(f( f( 4 ) )\)
118. \(f( f( 1 ) )\)
119. \(g( g( 2 ) )\)
120. \(g( g( 6 ) )\)

For the following exercises, use the function values for \(f\) and \(g\) shown in Table \( \PageIndex{ 2 } \) to evaluate the expressions.

121. \((f \circ g)(1)\)
122. \((f \circ g)(2)\)
123. \((g \circ f)(2)\)
124. \((g \circ f)(3)\)
125. \((g \circ g)(1)\)
126. \((f \circ f)(3)\)

For the following exercises, use each pair of functions to find \(f( g( 0 ) )\) and \(g( f(0) )\).

127. \(f(x)=4x+8\), \(g(x)=7− x^2\)
128. \(f(x)=5x+7\), \(g(x)=4−2 x^2\)
129. \(f(x)=\sqrt{ x+4 }\), \(g(x)=12− x^3\)
130. \(f(x)= \frac{1}{x+2}\), \(g(x)=4x+3\)

For the following exercises, use the functions \(f(x)=2 x^2 +1\) and \(g(x)=3x+5\) to evaluate or find the composite function as indicated.

131. \(f( g(2) )\)
132. \(f( g(x) )\)
133. \(g( f(−3) )\)
134. \((g \circ g)(x)\)

Extensions

For the following exercises, let \( f \) be the function defined by \[ f=\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\} \nonumber\] and let \( g\) be the function defined \[g=\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\} \nonumber \] Compute the indicated value if it exists.

135. \(\ (f + g)(−3)\)
136. \(\ (f − g)(2)\)
137. \(\ (fg)(−1)\)
138. \(\ (g + f)(1)\)
139. \(\ (g − f)(3)\)
140. \(\ (gf)(−3)\)
141. \(\ \left(\frac{f}{g}\right)(-2)\)
142. \(\ \left(\frac{f}{g}\right)(-1)\)
143. \(\ \left(\frac{f}{g}\right)(2)\)
144. \(\ \left(\frac{g}{f}\right)(-1)\)
145. \(\ \left(\frac{g}{f}\right)(3)\)
146. \(\ \left(\frac{g}{f}\right)(-3)\)

For the following exercises, use \(f(x)= x^3 +1\) and \(g(x)= \sqrt[3]{x−1} \).

147. Find \((f \circ g)(x)\) and \((g \circ f)(x)\). Compare the two answers.
148. Find \((f \circ g)(2)\) and \((g \circ f)(2)\).
149. What is the domain of \((g \circ f)(x)\)?
150. What is the domain of \((f \circ g)(x)\)?
151. Let \(f(x)= \frac{1}{x} \).
     1. Find \((f \circ f)(x)\).
     2. Is \((f \circ f)(x)\) for any function \(f\) the same result as the answer to part (a) for any function? Explain.

For the following exercises, let \(F(x)= (x+1)^5 \), \(f(x)= x^5 \), and \(g(x)=x+1\).

152. True or False: \((g \circ f)(x)=F(x)\).
153. True or False: \((f \circ g)(x)=F(x)\).

For the following exercises, find the composition when \(f(x)= x^2 +2\) for all \(x \geq 0\) and \(g(x)=\sqrt{ x−2 } \).

154. \((f \circ g)(6)\); \((g \circ f)(6)\)
155. \((g \circ f)(a)\); \((f \circ g)(a)\)
156. \((f \circ g)(11)\); \((g \circ f)(11)\)

Real-World Applications

157. The function \(D(p)\) gives the number of items that will be demanded when the price is \(p\). The production cost \(C(x)\) is the cost of producing \(x\) items. To determine the cost of production when the price is $6, you would do which of the following?
     1. Evaluate \(D( C(6) )\).
     2. Evaluate \(C( D(6) )\).
     3. Solve \(D( C(x) )=6\).
     4. Solve \(C( D(p) )=6\).
158. The function \(A(d)\) gives the pain level on a scale of 0 to 10 experienced by a patient with \(d\) milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after \(t\) minutes is modeled by \(m(t)\). Which of the following would you do in order to determine when the patient will be at a pain level of 4?
     1. Evaluate \(A( m(4) )\).
     2. Evaluate \(m( A(4) )\).
     3. Solve \(A( m(t) )=4\).
     4. Solve \(m( A(d) )=4\).
159. A store offers customers a 30% discount on the price \(x\) of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function \(P(x)\) that computes the final price of the item in terms of the original price \(x\). (Hint: Use function composition to find your answer.)
160. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to \(r(t)=25\sqrt{ t+2 } \), find the area of the ripple as a function of time. Find the area of the ripple at \(t=2\).
161. A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula \(r(t)=2t+1\), express the area burned as a function of time, \(t\) (minutes).
162. Use the function you found in the previous exercise to find the total area burned after 5 minutes.
163. The radius \(r\), in inches, of a spherical balloon is related to the volume, \(V\), by \( r(V)= \sqrt[3]{\frac{3V}{4 \pi}}\). Air is pumped into the balloon, so the volume after \(t\) seconds is given by \( V(t)=10+20t\).
     1. Find the composite function \(r( V(t) )\).
     2. Find the exact time when the radius reaches 10 inches.
164. The number of bacteria in a refrigerated food product is given by \( N(T)=23 T^2 − 56T+1\), \(3<T<33\), where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by \(T(t)=5t+1.5\), where \(t\) is the time in hours.
     1. Find the composite function \(N( T(t) )\).
     2. Find the time (round to two decimal places) when the bacteria count reaches 6752.

For the following exercises, \(\ C(x)\) denotes the cost to produce \(\ x\) items and \(\ p(x)\) denotes the price-demand function in the given economic scenario. In each Exercise, do the following:

• Find and interpret \(\ C(0)\).
• Find and interpret \(\ \overline{C}(10)\). (\( \overline{C} \) is called the average cost)
• Find and interpret \(\ p(5)\).
• Find and simplify \(\ R(x)\). (\( R(x) \) is the revenue function)
• Find and simplify \(\ P(x)\). (\( P(x) \) is the profit function)
• Solve \(\ P(x) = 0\) and interpret.

165. The cost, in dollars, to produce \(\ x\) "I’d rather be a Sasquatch" T-Shirts is \(\ C(x) = 2x + 26\), \(\ x \geq 0\) and the price-demand function, in dollars per shirt, is \(\ p(x) = 30 − 2x\), \(\ 0 \leq x \leq 15\).
166. The cost, in dollars, to produce \(\ x\) bottles of 100% All-Natural Certified Free-Trade Organic Sasquatch Tonic is \(\ C(x) = 10x + 100\), \(\ x \geq 0\) and the price-demand function, in dollars per bottle, is \(\ p(x) = 35 − x, 0 \leq x \leq 35\).
167. The cost, in cents, to produce x cups of Mountain Thunder Lemonade at Junior’s Lemonade Stand is \(\ C(x) = 18x + 240\), \(\ x \geq 0\) and the price-demand function, in cents per cup, is \(\ p(x) = 90 − 3x, 0 \leq x \leq 30\).
168. The daily cost, in dollars, to produce \(\ x\) Sasquatch Berry Pies \(\ C(x) = 3x + 36\), \(\ x \geq 0\) and the price-demand function, in dollars per pie, is \(\ p(x) = 12 − 0.5x\), \(\ 0 \leq x \leq 24\).
169. The monthly cost, in hundreds of dollars, to produce \( x \) custom built electric scooters is \(\ C(x) = 20x + 1000\), \(\ x \geq 0\) and the price-demand function, in hundreds of dollars per scooter, is \(\ p(x) = 140 − 2x\), \(\ 0 \leq x \leq 70\).