1.7E: Inverse Functions (Exercises)
- Page ID
- 192363
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Assume that the inverse of function f is also a function.
- If \(f(6)=7\), find \(f^{-1} (7)\)
- If \(f(3)=2\), find \(f^{-1} (2)\)
- If \(f^{-1} (-4)=-8\), find \(f(-8)\)
- If \(f^{-1} (-2)=-1\), find \(f(-1)\)
2. If the function \(f\) is given in the table below, find a table that defines the inverse function \(f^{-1}\).
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(y=f(x)\) | 7 | 1 | 3 | 5 | 6 |
3. Use the table below to find the indicated quantities.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| \(g(x)\) | 8 | 0 | 7 | 4 | 2 | 6 | 5 | 3 | 9 | 1 |
- Find \(g(0)\)
- Find \(g^{-1}(0)\)
- Find all values of x such that \(g^{-1}(x)=7\)
- Write a table of values that defines the inverse function \(g^{-1}\)
4. The function \(f\) is given by the set of ordered pairs \( f \) = {(-2,3), (3,1), (1,5), (2,4)}. Find a set of ordered pairs that defines the inverse function \(f^{-1}\).
5. Using the graph of \(f(x)\) shown
- Find \(f(0)\)
- Find \(f^{-1} (0)\)
- Find \(f^{-1} (1)\)
- Find all values of \(y\) such that \(f^{-1} (y)=0\)
6. Using the graph shown
- Find \(g(1)\)
- Find \(g^{-1} (1)\)
- Find \(g^{-1} (2)\)
- Find all values of \(y\) such that \(g^{-1} (y)=3\)
7. Find a formula for the inverse \(f^{-1}\) of each function
- \(f(x)=2x+3\)
- \(f(x)=(x +7)^{2}\)
- \(f(x)=\sqrt[3]{3x-7}\)
8. Find a formula for the inverse \(g^{-1}\) of each function
- \(g(x)=5x-8\)
- \(g(x)=(3x +7)^{4}\)
- \(g(x)=2+ \sqrt{6x-1}\)
9. Find a formula for the inverse \(f^{-1}\) of each function
- \(f(x)=\dfrac{x+1}{x-2}\)
- \(f(x)=\dfrac{3x-2}{x+7}\)
10. 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 \(g(x)\)?
11. 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 \(g(x)\)?
12. Determine which of the functions are inverses of each other where \(f(x)=5x^2+3\), \(g(x)=\dfrac{\sqrt{x}-3}{5}\), and \(h(x) =\sqrt{ \dfrac{x-3}{5}}\).
13. The function \(f\) is given by the set of ordered pairs \( f \)= {(0,1), (1,-2), (2,0), (3,2) (4,1)}.
- In the inverse \(f^{-1}\) a function for all defined values of x?
- If you answered "Yes, \(f^{-1}\) is a function" in part (a), find a new ordered pair that you can add to function \(f\) that would make \(f^{-1}\) not a function. If you answered "No, \(f^{-1}\) is not a function" in part (a), find an ordered pair(s) that you can remove from the function \(f\) that would make \(f^{-1}\) a function.
14. The function \(g\) is given by the set of ordered pairs \( g \)= {(1,2), (3,4), (2,3), (5,1) (6,5)}.
- In the inverse \(g^{-1}\) a function for all defined values of x?
- If you answered "Yes, \(g^{-1}\) is a function" in part (a), find a new ordered pair that you can add to function \(g\) that would make \(g^{-1}\) not a function. If you answered "No, \(g^{-1}\) is not a function" in part (a), find an ordered pair(s) that you can remove from the function \(g\) that would make \(g^{-1}\) a function.
15. Let \(f(x)=x^2(x-2)^2\). Graph \(f(x)\) with your graphing utility. Is the inverse \(f^{-1}\) a function for all values of x? If not, how can you restrict the domain of f so that \(f^{-1}\) is a function.
16. The graph of \(g(x)=x^3 +7x+2\) is given below. Is the inverse \(g^{-1}\) a function for all values of x? If not, how can you restrict the domain of g so that \(g^{-1}\) is a function.
17. The graph of \(f(x) = 3cos(0.5 \pi x)\) is given below. Is the inverse \(f^{-1}\) a function for all values of x? If not, how can you restrict the domain of f so that \(f^{-1}\) is a function.
18. If \(T = f(H)\) represents the time \(T\) in minutes it takes to bake a pizza at a temperature \(H\) in degrees Fahrenheit. Write a sentence that interpret the meaning of the following statements.
- \( f(400)= 12\)
- \( f^{-1}(10)= 450\)
19.If \(V = g(d)\) represents the volume of water \(V\) in cubic feet that will flow through a pipe with interior diameter \(d\) in inches in one minute. Write a sentence that interpret the meaning of the following statements.
- \( g(1)= 6.4\)
- \( g^{-1}(6.4)= 20.2\)
20. The cost (in dollars) of producing \(x\) units of a product is given by \( C = f(x)=24x+700\) .
- Evaluate and interpret \( f(4000)\)
- Evaluate and interpret \( f^{-1}(4000)\)
21. The stopping distance of a car (in feet) traveling at an initial velocity of \(v\) mph is given by \( D = g(v)=\dfrac{v^2}{8}\) .
- Evaluate and interpret \( g(60)\)
- Evaluate and interpret \( g^{-1}(60)\)
22. The average value of a solar panel in the US (in dollars per peak Watt) \(t\) years since 2012 can estimated with the function \( V = f(t)=0.01(t-9)^2+0.34\) using data from The US Energy Information Administration.
- Evaluate and interpret \( f(6)\)
- Evaluate and interpret \( f^{-1}(0.87)\)
- Answer
-
1. a) 6 c) -4
3. a) 8 b ) 1 c) 3 d) shown in the table below
\(y\) 8 0 7 4 2 6 5 3 9 1 \(x=f^{-1}(y)\) 0 1 2 3 4 5 6 7 8 9 5. a) 3 b) 2 c) \(y\approx 1.3\) d) 3
7. a) \( x = f^{-1}(y) = \dfrac{y+3}{2} \) c) \( x = f^{-1}(y) = \dfrac{y^3+7}{3} \)
9. a) \( x = f^{-1}(y) = \dfrac{y+1}{y-1} \)
11. a) x b) x c) They are inverses.
13. a) \(f^{-1} \) is a function b) Adding \(f^{-1} \) would make it not a function.
17. a) At 400 degrees, it takes 12 minutes to cook the pizza. b) At 450 degrees, it takes 10 minutes to cook the pizza.
20. a) The cost of producing 4000 units is $96,100. b) The cost of producing 137.5 units is $4,000.