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: 1


The required values are $f\left( g\left( x \right) \right)=x$ and $g\left( f\left( x \right) \right)=x$. The functions $f\left( x \right)=4x$and $g\left( x \right)=\frac{x}{4}$ are inverses of each other.

Work Step by Step

Consider the functions: $f\left( x \right)=4x$ and $g\left( x \right)=\frac{x}{4}$ The equation for $f$ is given as: $f\left( x \right)=4x$ Replace $x$ with $g\left( x \right)$ $\begin{align} & f\left( g\left( x \right) \right)=4g\left( x \right) \\ & =4\left( \frac{x}{4} \right) \\ & =x \end{align}$ Now, to find $g\left( f\left( x \right) \right)$ Consider the function $g\left( x \right)$: $g\left( x \right)=\frac{x}{4}$ Replace $x$ with $f\left( x \right)$ $\begin{align} & g\left( f\left( x \right) \right)=\frac{f\left( x \right)}{4} \\ & =\frac{\left( 4x \right)}{4} \\ & =x \end{align}$ Because $g$ is the inverse of $f$ (and vice-versa), the inverse notation can be used: $f\left( x \right)=4x$ and $\text{ }{{f}^{-1}}\left( x \right)=\frac{x}{4}$ It can be easily observed that ${{f}^{-1}}$ can be expressed in $f$ if multiplied by 4. $\begin{align} & \text{ }{{f}^{-1}}\left( x \right)=\frac{x}{4} \\ & =g\left( x \right) \end{align}$ Hence, the values are $f\left( g\left( x \right) \right)=x$ and $g\left( f\left( x \right) \right)=x$, the functions \[f\left( x \right)\] and \[g\left( x \right)\]are inverse of each other.
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