## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 1 - Section 1.8 - Inverse Functions - Exercise Set - Page 269: 24

#### Answer

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

#### Work Step by Step

(a) Consider the function: $f\left( x \right)=\sqrt[3]{x}$ Follow the procedure to determine the inverse ${{f}^{-1}}\left( x \right)$ of the function $f\left( x \right)=\sqrt{x}$. Step 1: Replace the function $f\left( x \right)$ with y in the equation of $f\left( x \right)$. $y=\sqrt[3]{x}$ Step 2: Interchange the variables x and y. $x=\sqrt[3]{y}$ Step 3: Solve the equation for y. The variable y has to be isolated. So cube both sides of the equation. So, the equation becomes, \begin{align} & {{\left( x \right)}^{3}}={{\left( \sqrt[3]{y} \right)}^{3}} \\ & {{x}^{3}}=y \end{align} Because the obtained equation defines y as a function of x for all $x\ge 0$ , the inverse function exists for the function f. Step 4: Replace y by ${{f}^{-1}}\left( x \right)$ in the equation obtained in step 3. Thus, the inverse function${{f}^{-1}}\left( x \right)$ is obtained as, ${{f}^{-1}}\left( x \right)={{x}^{3}}$ Therefore, the inverse function is ${{f}^{-1}}\left( x \right)={{x}^{3}}$. (b) Consider the function: $f\left( x \right)=\sqrt[3]{x}$. The inverse of the function is obtained as ${{f}^{-1}}\left( x \right)={{x}^{3}}$ in part (a). The inverse of an equation can be verified by showing that $f\left( {{f}^{-1}}\left( x \right) \right)=x\text{ and }{{f}^{-1}}\left( f\left( x \right) \right)=x\text{ }$. First calculate $f\left( {{f}^{-1}}\left( x \right) \right)$. Replace ${{f}^{-1}}\left( x \right)={{x}^{3}}$ and evaluate the function f. \begin{align} & f\left( {{f}^{-1}}\left( x \right) \right)=f\left( {{x}^{3}} \right) \\ & =\sqrt[3]{{{x}^{3}}} \\ & =x \end{align} Now, calculate ${{f}^{-1}}\left( f\left( x \right) \right)$. Replace $f\left( x \right)=\sqrt[3]{x}$ and evaluate the inverse function ${{f}^{-1}}$. \begin{align} & {{f}^{-1}}\left( f\left( x \right) \right)={{f}^{-1}}\left( \sqrt[3]{x} \right) \\ & ={{\left( \sqrt[3]{x} \right)}^{3}} \\ & =x \end{align} Thus, it verifies that $f\left( {{f}^{-1}}\left( x \right) \right)=x\text{ and }{{f}^{-1}}\left( f\left( x \right) \right)=x\text{ }$. Therefore, the obtained inverse function ${{f}^{-1}}\left( x \right)={{x}^{3}}$ is correct. Hence, the functions $f$ and ${{f}^{-1}}$ are inverses of each other.

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