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Section 8
Inverse Functions
Taylor said that if $(a, b)$ is a pair of a one-to-one function $f,$ then $(b, a)$ must be a pair of the inverse function $f^{-1} .$ Do you agree with Taylor? Explain why or why not.
Christopher said that $\mathrm{f}(x)=|x-2|$ and $\mathrm{g}(x)=|x+2|$ are inverse functions after he showed that $\mathrm{f}(\mathrm{g}(2))=2, \mathrm{f}(\mathrm{g}(5))=5,$ and $\mathrm{f}(\mathrm{g}(7))=7 .$ Do you agree that $\mathrm{f}$ and $\mathrm{g}$ are inverse functions? Explain why or why not.
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{I}(3)$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{I}(5)$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{I}(-2)$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{I}(\mathrm{f}(2))$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{f}(\mathrm{I}(3))$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{f}\left(\mathrm{f}^{-1}(-6)\right)$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{f}^{-1}(\mathrm{f}(-6))$$
In $3-10,$ find each of the function values when $\mathrm{f}(x)=4 x$$$\mathrm{f}\left(\mathrm{f}^{-1}(\sqrt{2})\right)$$
In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\{(0,8),(1,7),(2,6),(3,5),(4,4)\}$$
In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\{(1,4),(2,7),(1,10),(4,13)\}$$
In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\{(0,8),(2,6),(4,4),(6,2)$$
In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\{(2,7),(3,7),(4,7),(5,7),(6,7)\}$$
In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\{(-1,3),(-1,5),(-2,7),(-3,9),(-4,11)\}$$
In $11-16,$ determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function.$$\left\{(x, y) : y=x^{2}+2 \text { for } 0 \leq x \leq 5\right\}$$
In $17-20 :$ a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers.$$f(x)=4 x-3$$
In $17-20 :$ a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers.$$g(x)=x-5$$
In $17-20 :$ a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers.$$f(x)=\frac{x+5}{3}$$
In $17-20 :$ a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers.$$\mathrm{f}(x)=\sqrt{x}$$
If $\mathrm{f}=\{(x, y) : y=5 x\}$ is a direct variation function, find $\mathrm{f}^{-1}$
If $\mathrm{g}=\{(x, y) : y=7-x\},$ find $\mathrm{g}^{-1}$ if it exists. Is it possible for a function to be its own inverse?
Does $y=x^{2}$ have an inverse function if the domain is the set of real numbers? Justify your answer.
In $24-26,$ sketch the inverse of the given function.(SKETCH NOT COPY)
On a particular day, the function that converts American dollars, $x,$ to Indian rupees, $f(x),$ is$\mathrm{f}(x)=0.2532 x .$ Find the inverse function that converts rupees to dollars. Verify that thefunctions are inverses.
When the function $g(x)=x^{2}+8 x+18$ is restricted to the interval $x \geq 2,$ the inverse is$g^{-1}(x)=\sqrt{x-2}-4$a. Graph g for values of $x \geq 2 .$ Graph $\mathrm{g}^{-1}$ on the same set of axes.b. What is the domain of $\mathrm{g}$ ? What is the range of $\mathrm{g}$ ?c. What is the domain of $\mathrm{g}^{-1}$ ? What is the range of $\mathrm{g}^{-1}$ ?d. Describe the relationship between the domain and range of $\mathrm{g}$ and its inverse.