1.6E: Inverse Functions
- Page ID
- 30243
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)section 1.6 exercises
Assume that the function f is a one-to-one function.
1. If \(f(6)=7\), find \(f^{-1} (7)\)
2. If \(f(3)=2\), find \(f^{-1} (2)\)
3. If \(f^{-1} (-4)=-8\), find \(f(-8)\)
4. If \(f^{-1} (-2)=-1\), find \(f(-1)\)
5. If \(f(5)=2\), find \((f(5))^{-1}\)
6. If \(f(1)=4\), find \((f(1))^{-1}\)
7. Using the graph of \(f(x)\) shown
a. Find \(f(0)\)
b. Solve \(f(x)=0\)
c. Find \(f^{-1} (0)\)
d. Solve \(f^{-1} (x)=0\)
8. Using the graph shown
a. Find \(g(1)\)
b. Solve \(g(x)=1\)
c. Find \(g^{-1} (1)\)
d. Solve \(g^{-1} (x)=1\)
9. Use the table below to find the indicated quantities.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| \(f(x)\) | 8 | 0 | 7 | 4 | 2 | 6 | 5 | 3 | 9 | 1 |
a. Find \(f(1)\)
b. Solve \(f(x)=3\)
c. Find \(f^{-1}(0)\)
d. Solve \(f^{-1}(x)=7\)
10. Use the table below to fill in the missing values.
| \(t\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| \(h(t)\) | 6 | 0 | 1 | 7 | 2 | 3 | 5 | 4 | 9 |
a. Find \(h(6)\)
b. Solve \(h(t)=0\)
c. Find \(h^{-1} (5)\)
d. Solve \(h^{-1} (t)=1\)
For each table below, create a table for \(f^{-1} (x).\)
11.
| \(x\) | 3 | 6 | 9 | 13 | 14 |
| \(f(x)\) | 1 | 4 | 7 | 12 | 16 |
For each function below, find \(f^{-1} (x)\)
13. \(f(x)=x+3\)
14. \(f(x)=x+5\)
15. \(f(x)= 2 - x\)
16. \(f(x)=3-x\)
17. \(f(x)=11x+7\)
18. \(f(x)=9+10x\)
For each function, find a domain on which \(f\) is one-to-one and non-decreasing, then find the inverse of \(f\) restricted to that domain.
19. \(f(x)=(x +7)^{2}\)
20. \(f(x)=(x-6)^{2}\)
21. \(f(x)=x^{2} -5\)
22. \(f(x)=x^{2} +1\)
23. If \(f(x)=x^{3} -5\) and \(g(x)=\sqrt[{3}]{x+5}\), find
a. \(f(g(x))\)
b. \(g(f(x))\)
c. What does this tell us about the relationship between \(f(x)\) and \(g(x)\)?
24. If \(f(x)=\dfrac{x}{2+x}\) and \(g(x)=\dfrac{2x}{1-x}\), find
a. \(f(g(x))\)
b. \(g(f(x))\)
c. What does this tell us about the relationship between \(f(x)\) and \(g(x)\)?
- Answer
-
1. 6
3. -4
5. 1/2
7a. 3
b. 2
c. 2
d. 211.
\(x\) 1 4 7 12 16 \(f^{-1}(x)\) 3 6 9 13 14 13. \(f^{-1}(x) = x -3\)
15. \(f^{-1}(x) = -x + 2\)
17. \(f^{-1}(x) = \dfrac{x - 7}{11}\)
19. Restricted domain \(x \ge -7\), \(f^{-1}(x) = \sqrt{x} - 7\)
21. Restricted domain \(x \ge 0\), \(f^{-1}(x) = \sqrt{x + 5}\)
23a. \(f(g(x)) = (\sqrt[3]{x + 5})^3 - 5 = x\)
b. \(g(f(x)) = \sqrt[3]{x^3 - 5 + 5} = x\)
c. This means that they are inverse functions (of each other)