## Algebra and Trigonometry 10th Edition

Published by Cengage Learning

# Chapter 2 - 2.7 - Inverse Functions - 2.7 Exercises - Page 229: 48

#### Answer

a) $f^{-1}(x)=-\sqrt{x+2}$ b) See graph c) The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. d) $D_f=(-\infty,0],R_f=[-2,\infty)$ $D_{f^{-1}}=[-2,\infty),R_{f^{-1}}=(-\infty,0]$

#### Work Step by Step

We are given the function: $f(x)=x^2-2,x\in (-\infty,0]$ $y=x^2-2$ a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$: $x=y^2-2,y\in (-\infty,0]$ $y^2=x+2$ $y=-\sqrt{x+2}$ because $y\in (-\infty,0]$ $f^{-1}(x)=-\sqrt{x+2}$ 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 $f$: $D_f=(-\infty,0]$ $R_f=[-2,\infty)$ Determine the domain and range of $f^{-1}$: $D_{f^{-1}}=[-2,\infty)$ $R_{f^{-1}}=(-\infty,0]$

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