Chapter 1.9, Problem 68E

To determine

Whether the given function $f(x)=\left|x-2\right|,x\le 2$ has an inverse function and if it does, find the inverse function.

Answer to Problem 68E

The inverse function of given function is $f^{-1}(x)=-x+2,x\le 2$

Explanation of Solution

Given information :

Given function $f(x)=\left|x-2\right|,x\le 2$

A function $f(x)$ has an inverse function, if it is a one-one function.

Since $f(x)=\left|x-2\right|,x\le 2$ is one-one as each value of dependent variable corresponds to exactly one value of the independent variable.

Thus, the function $f(x)=\left|x-2\right|,x\le 2$ has an inverse function.

To find inverse function:

Step 1: Replace $y=f(x)$ in $f(x)=\left|x-2\right|$ , we get

  $y=\left|x-2\right|$

Step 2: interchange role of $x\text{and}y$ . And solve for $y$ .

  $\begin{array}{l}x=\left|y-2\right|\\ \Rightarrow -x=\left(y-2\right),x\le 2\\ \Rightarrow -x=y-2\\ \therefore y=-x+2,x\le 2\end{array}$

Step 3: replace $y$ with $f^{-1}(x)$ in new equation, we get

  $f^{-1}(x)=-x+2,x\le 2$

Therefore, the inverse function of given function is $f^{-1}(x)=-x+2,x\le 2$