# 2.5e: Exercises Inverse Functions

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

### A: Concepts

Exercise $$\PageIndex{A}$$

1) Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

2) Why do we restrict the domain of the function $$f(x)=x^2$$ to find the function’s inverse?

3) Can a function be its own inverse? Explain.

4) Are one-to-one functions either always increasing or always decreasing? Why or why not?

5) How do you find the inverse of a function algebraically?

1. Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that $$y$$-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no $$y$$-values repeat and the function is one-to-one.

3. Yes. For example, $$f(x)=\dfrac{1}{x}$$ is its own inverse.

5. Given a function $$y=f(x)$$, solve for $$x$$ in terms of $$y$$. Interchange the $$x$$ and $$y$$. Solve the new equation for $$y$$. The expression for $$y$$ is the inverse, $$y=f^{-1}(x)$$.

### B: Horizontal Line Test

Exercise $$\PageIndex{B}$$)

$$\bigstar$$ For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

7. Not one-to-one, $$\;$$ 9. Not one-to-one, $$\;$$ 11. One-to-one, $$\;$$ 13. Not one-to-one, $$\;$$ 15. One-to-one, $$\;$$ 17. Not one-to-one

### C: Graphs of Inverse Functions

Exercise $$\PageIndex{C}$$

$$\bigstar$$ For the following exercises, use the graph of $$f$$ to sketch the graph of its inverse function.

 19) 20) 21) 22)

$$\bigstar$$ Use the graph of the one-to-one function shown in the Figure to answer the following questions.

 23) Find $$f(0) \\[4pt]$$. 24) Solve $$f(x)=0 \\[4pt]$$. 25) Find $$f^{-1}(0) \\[4pt]$$. 26) Solve $$f^{-1}(x)=0$$. 27) Sketch the graph of $$f^{-1} \\[4pt]$$. 28) Find $$f(6)$$ and $$f^{-1}(2) \\[4pt]$$. 29) If the complete graph of $$f$$ is shown, find the domain of $$f \\[4pt]$$. 30) If the complete graph of $$f$$ is shown, find the range of $$f$$
 19) 21) $$\;$$ 23. $$3$$  25. $$2$$ 29. $$[2, \infty )$$ 27

### D: Inverse Function Values

Exercise $$\PageIndex{D}$$

$$\bigstar$$ Find values of a function from its inverse and vice versa.

 31) If $$f(6)=7$$, find $$f^{-1}(7)$$. 32) If $$f(3)=2$$, find $$f^{-1}(2)$$. 32.1) If $$f(4)=5$$, find $$f^{-1}(4)$$. 32.2) If $$f^{-1}(9)=3$$, find $$f(9)$$. 33) If $$f^{-1}(−4)=−8$$, find $$f(−8)$$. 34) If $$f^{-1}(−2)$$=−1, find $$f(−1)$$.

$$\bigstar$$ Given the table of values for $$f$$, determine the following.

$$\begin{array}{ll} 35) \text{ Find } f(1). \qquad \qquad & 36) \text{ Solve} f(x)=3. \\ 37) \text{ Find } f^{-1}(4).& 38) \text{ Solve } f^{-1}(x)=7. \\ 39) \text{ Solve } f^{-1}(x)=11. & 40) \text{ Find } f^{-1}(12). \end{array}$$

 $$x$$ $$f(x)$$ 1 4 7 12 16 12 11 9 4 3

31. $$6$$ $$\qquad$$ 33. $$-4$$, $$\qquad$$ 35. $$12$$, $$\qquad$$ 37. $$12$$, $$\qquad$$ 39. Undefined

### E: Verify Two Functions are Inverses

Exercise $$\PageIndex{E}$$

$$\bigstar$$ For the following exercises, use composition to determine which pairs of functions are inverses.

 41) $$f(x)=8x, \; g(x)=\dfrac{x}{8}$$ 42) $$f(x)=\dfrac{2}{3}x+2, \; g(x)=\dfrac{3}{2}x+3$$ 43) $$f(x)=5x−7, \; g(x)=\dfrac{x+5}{7}$$ 44) $$f(x)=8x+3, \; g(x)=\dfrac{x-3}{8}$$ 45) $$f(x)=\dfrac{1}{x−1}, x≠1, \; g(x)=\dfrac{1}{x}+1,x≠0$$ 46) $$f(x)=−3x+5$$,    $$g(x)=\dfrac{x-5}{-3}$$ 47) $$f(x)=\dfrac{x}{2+x}$$,  $$g(x)=\dfrac{2x}{1-x} \\[2pt]$$ 48) $$f(x)=x^3+1, \; g(x)=(x−1)^{1/3} \\[2pt]$$ 49) $$f(x)=x^2+2x+1,x≥−1, \; g(x)=−1+\sqrt{x},x≥0 \\[2pt]$$ 50) $$f(x)=\sqrt{4−x^2},0≤x≤2, \; g(x)=\sqrt{4−x^2},0≤x≤2 \\[2pt]$$

41) inverses.     43) not inverses.    45) inverses.      47) inverses    49) inverses

### F: Find inverses of linear and rational functions

Exercise $$\PageIndex{F}$$

$$\bigstar$$ Find the formula for the inverse function $$f^{−1}(x)$$.

 51) $$f(x)=x+3 \\[2pt]$$ 52) $$f(x)=x+5 \\[2pt]$$ 53) $$f(x)=2−x \\[2pt]$$ 54) $$f(x)=3−x \\[2pt]$$ 55) $$f(x) = 7x−9 \\[2pt]$$ 56) $$f(x) = 6x−4 \\[2pt]$$ 57) $$f(x) = −5x+2 \\[2pt]$$ 58) $$f(x) = 6x+8$$ 59) $$f(x)=\dfrac{3}{x+2} \\[3pt]$$ 60) $$f(x)=\dfrac{x}{x-2} \\[3pt]$$ 61) $$f(x)= \dfrac{x-4}{x+2} \\[3pt]$$ 62) $$f(x)= \dfrac{2x-7}{x+6} \\[3pt]$$ 63) $$f(x)= \dfrac{2x+3}{5x+4} \\[3pt]$$ 64) $$f(x)= \dfrac{2x+6}{x-3} \\[3pt]$$ 65) $$f(x) = \dfrac{9x−3}{9x+7}$$ 66)  $$f(x) = \dfrac{3x+7}{2x+8} \\[3pt]$$ 67)  $$f(x) = \dfrac{4x+2}{4x+3} \\[3pt]$$ 68)  $$f(x) = \dfrac{8x−7}{3x−6} \\[3pt]$$ 69)  $$f(x) = \dfrac{4x−1}{2x+2} \\[3pt]$$ 70) $$f(x)=−\dfrac{9x−3}{7x+6}$$

51. $$f^{-1}(x)=x−3$$     53. $$f^{-1}(x)=2−x$$
55. $$f^{-1}(x)=\frac{x+9}{7}$$     57. $$f^{-1}(x)=−\frac{x−2}{5}$$    59. $$f^{-1}(x) = \frac{3}{x} - 2$$     61. $$f^{-1}(x) = -2\frac{x+2}{x-1}$$
63. $$f^{-1}(x) = \frac{-4x+3}{5x-2}$$     65. $$f^{-1}(x)=−\frac{7x+3}{9x−9}$$     67. $$f^{-1}(x)=−\frac{3x−2}{4x−4}$$     69. $$f^{-1}(x)=−\frac{2x+1}{2x−4}$$

### G: Find inverses of odd degree power and root functions

Exercise $$\PageIndex{G}$$

$$\bigstar$$ Find the formula for the inverse function $$f^{−1}(x)$$.

 71) $$f(x)=x^3+1 \\[2pt]$$ 72) $$f(x)=x^3−27 \\[2pt]$$ 73) $$f(x)=(x-4)^3 \\[2pt]$$ 74) $$y=(x+8)^3+3$$ 75)  $$f(x) = 5x^3−5 \\[2pt]$$ 76) $$f(x) = 4x^5−9 \\[2pt]$$ 77) $$f(x) = 3x^5−9 \\[2pt]$$ 78) $$f(x) = 5x^7+4$$ 79) $$f(x) = 9x^9+5 \\[2pt]$$ 80) $$f(x) = 4x^7−3 \\[2pt]$$ 81) $$f(x)=\sqrt[3]{x−4} \\[2pt]$$ 82) $$f(x)=\sqrt[3]{3x+1}$$ 83) $$f(x)=\sqrt[3]{x} + 5 \\[2pt]$$ 84) $$f(x)=\sqrt[3]{x} - 8 \\[2pt]$$ 85) $$f(x) = \sqrt[3]{−6x−4} \\[2pt]$$ 86) $$f(x) = \sqrt[3]{9x−7}$$ 87) $$f(x) = \sqrt[7]{−3x−5} \\[2pt]$$ 88) $$f(x) = \sqrt[7]{8x−3} \\[2pt]$$ 89) $$f(x) = \sqrt[3]{6x+7} \\[2pt]$$ 90) $$f(x) = \sqrt[9]{8x+2}$$

71. $$f^{−1}(x)=\sqrt[3]{x-1}$$     73. $$f^{−1}(x) = 4 + \sqrt[3]{x}$$     75. $$\sqrt[3]{\frac{x+5}{5}}$$     77. $$\sqrt[5]{\frac{x+9}{3}}$$     79. $$\sqrt[9]{\frac{x−5}{9}}$$
81. $$f^{-1}(x) = x^3+4$$     83. $$f^{-1}(x) = (x-5)^3$$     85. $$−\frac{x^3+4}{6}$$     87. $$−\frac{x^7+5}{3}$$     89. $$\frac{x^3−7}{6}$$

### H: Find inverses of even degree power and root functions

Exercise $$\PageIndex{H}$$ ]

$$\bigstar$$ Find $$f^{-1}(x)$$ for each function below. In #105-108 also state the restrictions for $$x$$  in  $$f^{-1}$$.

 91) $$f(x) = x^4, \;$$ $$x \le 0 \\[2pt]$$ 92) $$f(x) = x^4, \;$$ $$x \ge 0 \\[2pt]$$ 93) $$f(x) = x^2−1, \;$$ $$x \le 0 \\[2pt]$$ 94) $$f(x) = x^2+2, \;$$ $$x \ge 0 \\[2pt]$$ 95) $$f(x)=x^2−4, \; x≥0 \\[2pt]$$ 96) $$f(x)=x^2+11, \;x≤0$$ 97) $$f(x) = x^4+3, \;$$ $$x \le 0 \\[2pt]$$ 98) $$f(x) = x^4−5, \;$$ $$x \ge 0 \\[2pt]$$ 99) $$f(x)=(x−1)^2, \; x≥1 \\[2pt]$$ 100) $$f(x)=(x+3)^2, \; x≥-3 \\[2pt]$$ 101) $$f(x) = (x−1)^2, \;$$ $$x \le 1 \\[2pt]$$ 102) $$f(x) = (x+2)^2, \;$$ $$x \ge −2$$​​​​ 103) $$f(x)=x^2-8x+3, \; x≤4 \\[2pt]$$ 104) $$f(x)=x^2+2x+50, \; x≥-1 \\[2pt]$$ 105) $$f(x)=\sqrt{x−1} \\[2pt]$$ 106) $$f(x)=\sqrt{x+2} \\[2pt]$$ 107) $$f(x)=\sqrt{x}+9 \\[2pt]$$ 108) $$f(x)=\sqrt{x}-1$$
 91. $$f^{-1}(x) =−\sqrt[4]{x}$$ 93. $$f^{-1}(x) =−\sqrt{x+1}$$ 95. $$f^{-1}(x) = \sqrt{x+4}$$ 97. $$f^{-1}(x) =−\sqrt[4]{x−3}$$ 99. $$f^{-1}(x) = 1+ \sqrt{x}$$ 101. $$f^{-1}(x) =−\sqrt{x}+1$$ 103. $$f^{-1}(x) = 4-\sqrt{x+13}$$ 105.  $$f^{−1}(x)=x^2+1$$, $$x≥0$$ 107. $$f^{-1}(x) = (x-9)^2,$$  $$x≥9$$

### I: Find the inverse and its domain and range

Exercise $$\PageIndex{I}$$

$$\bigstar$$ Find a domain on which $$f$$ is one-to-one and non-decreasing.  Then find the inverse of $$f$$.

 111) $$f(x)=(x+7)^2$$ 112) $$f(x)=(x−6)^2$$ 113) $$f(x)=x^2−5$$ 114) $$f(x)=3(x−4)^2+1$$
111. domain of $$f(x)$$: $$\left[−7,\infty\right)$$; $$\quad$$ $$f^{-1}(x)=\sqrt{x}−7$$     113. domain of $$f(x)$$: $$\left[0,\infty\right)$$; $$\quad$$ $$f^{-1}(x)=\sqrt{x+5}$$