Chapter 1.9, Problem 75E

To determine

To find: inverse of the function f and then to state the domains of the function f and its

inverse.

Answer to Problem 75E

The inverse function $f^{-1}(x)=\{\begin{array}{l}x-2;x\ge -2\\ -x-2;x\le -2\end{array}$

Explanation of Solution

Given information:

The function $f(x)=|x+2|$

Concept used:

The function has inverse function if and only if the function is one to one.

Calculation.:

The function $f(x)=|x+2|$ is not one to one function.

To make one to one function

  $f(x)=x+2;x\ge -2$

Let $y=x+2$

  $\begin{array}{l}x=y-2\\ f^{-1}(y)=y-2\\ f^{-1}(x)=x-2\end{array}$

Or

  $f(x)=-(x+2);x\le -2$

Let $y=-(x+2)$

  $\begin{array}{l}x=-y-2\\ f^{-1}(y)=-y-2\\ f^{-1}(x)=-x-2\end{array}$

Hence, the inverse of the given function is $f^{-1}(x)=\{\begin{array}{l}x-2;x\ge -2\\ -x-2;x\le -2\end{array}$