Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 1 - Section 1.8 - Inverse Functions - Exercise Set - Page 269: 11


a) ${{f}^{-1}}\left( x \right)=x-3$ b) they are inverses

Work Step by Step

(a) Consider the function, $f\left( x \right)=x+3$ Step-1- Replace $f\left( x \right)$ with $y$. $\begin{align} & f\left( x \right)=x+3 \\ & y=x+3 \end{align}$ Step-2- Interchange $x$ and $y$ $\begin{align} & y=x+3 \\ & x=y+3 \\ \end{align}$ Step-3-Solve for $y$. Subtract $3$ from each side $\begin{align} & x=y+3 \\ & x-3=y+3-3 \\ & x-3=y \end{align}$ Step-4-Replace $y$ in step 3 by ${{f}^{-1}}\left( x \right)$ $\begin{align} & y=x-3 \\ & {{f}^{-1}}\left( x \right)=x-3 \end{align}$ Therefore, the required inverse of the function is ${{f}^{-1}}\left( x \right)=x-3$ (b) Consider the functions: $f\left( x \right)=x+3$ and ${{f}^{-1}}\left( x \right)=x-3$ Replace $x$ with ${{f}^{-1}}\left( x \right)$ in \[f\left( x \right)\] $\begin{align} & f\left( {{f}^{-1}}\left( x \right) \right)={{f}^{-1}}\left( x \right)+3 \\ & =\left( x-3 \right)+3 \end{align}$ Simplify, $\begin{align} & f\left( g\left( x \right) \right)=\left( x-3 \right)+3 \\ & =x \end{align}$ Now, find ${{f}^{-1}}\left( f\left( x \right) \right)$ The function ${{f}^{-1}}\left( x \right)$ is given as: ${{f}^{-1}}\left( x \right)=x-3$ Replace $x$ with $f\left( x \right)$ $\begin{align} & {{f}^{-1}}\left( f\left( x \right) \right)=f\left( x \right)-3 \\ & =\left( x+3 \right)-3 \end{align}$ Simplify $\begin{align} & {{f}^{-1}}\left( f\left( x \right) \right)=\left( x+3 \right)-3 \\ & =x \end{align}$ Thus, $f\left( {{f}^{-1}}\left( x \right) \right)=x$ and ${{f}^{-1}}\left( f\left( x \right) \right)=x$ It can be easily observed that ${{f}^{-1}}$ can be expressed in $f$ if subtracted by 3. Hence, the functions $f$ and \[{{f}^{-1}}\] are inverses of each other.
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