Q. 87 - Precalculus Enhanced with Graphing Utilities

Question

The function $f\left(x\right)=\left|x\right|$ is not one-to-one. Find a suitable restriction on the domain of $f$ so that the new function that results is one-to-one. Then find the inverse of $f$.

Step-by-Step Solution

Answer

The function and the inverse is

 $$\begin{gathered} f\left(x\right)=\left|x\right|,x\ge 0 \\ {f}^{-1}\left(x\right)=x,x\ge 0 \end{gathered}$$

Step 1: Given information

Given the function $f\left(x\right)=\left|x\right|$

Step 2: Calculating the inverse

The function is $f\left(x\right)=\left|x\right|$. Replacing $f\left(x\right)$ with $y$, we get

$$\begin{gathered} f\left(x\right)=\left|x\right| \\ y=x \\ x=y \\ {f}^{-1}\left(x\right)=x(x\ge 0) \end{gathered}$$