## Algebra and Trigonometry 10th Edition

Published by Cengage Learning

# Chapter 2 - 2.7 - Inverse Functions - 2.7 Exercises - Page 229: 53

#### Answer

a) $f^{-1}(x)=x^3+1$ b) See graph c) The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. d) $D_f=\left(-\infty,\infty\right),R_f=\left(-\infty,\infty\right)$ $D_{f^{-1}}=\left(-\infty,\infty\right),R_{f^{-1}}=\left(-\infty,\infty\right)$

#### Work Step by Step

We are given the function: $f(x)=\sqrt[3]{x-1}$ $y=\sqrt[3]{x-1}$ a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$: $x=\sqrt[3]{y-1}$ $x^3=(\sqrt[3]{y-1})^3$ $x^3=y-1$ $y=x^3+1$ $f^{-1}(x)=x^3+1$ b) Graph both functions. c) The graph of the function $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. d) Determine the domain and range of $f$: $D_f=\left(-\infty,\infty\right)$ $R_f=\left(-\infty,\infty\right)$ Determine the domain and range of $f^{-1}$: $D_{f^{-1}}=\left(-\infty,\infty\right)$ $R_{f^{-1}}=\left(-\infty,\infty\right)$

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