Figure

EXAMPLE 6Finding the Inverse of a Domain-Restricted Function

Find the inverse of $f(x) = x^{2}$ if $x\geq 0$ .

The function $f(x) = x^{2}$ is not one-to-one (see Example 1(a)). However, by restricting the domain of $f$ to $x\geq 0$ , the new function $f$ is one-to-one, so $f^{-1}$ exists. To find $f^{-1}$ , follow the steps.

Step 1

$y=x^{2}$ , where

$x\geq 0$ .

Step 2 Interchange the variables

$x$ and

$y$ :

$x = y^{2}$ , where

$y\geq 0$ . This is the inverse function written implicitly.

Step 3 Solve for

$y$ :

$y=\sqrt{x}=f^{-1}(x)$ . (Since

$y\geq 0$ , only the principal square root is obtained.)

Step 4 Check that

$f^{-1}(x) = \sqrt{x}$ is the inverse function of

$f$ .

$\begin{array}{rcl@{\qquad}l} f^{-1}( f( x))~&=&\sqrt{f( x) }=\sqrt{x^{2}}=\vert x\vert =x, & \hbox{where }x\geq 0 \\[2pt] f( f^{-1}( x))~&=&[ f^{-1}( x) ] ^{2}=[ \sqrt{x}] ^{2}=x, & \hbox{where }x\geq 0 \end{array}$