Chapter 5.3, Problem 55E

To determine

To graph: the inverse of the given function.

Explanation of Solution

Given information:

  $f\left(x\right)=x^2-16$

Calculation:

The $f^{-1}\left(x\right)$ can be evaluated as follows:

  $\begin{array}{c}y=x^2-16\left[\text{Let }f\left(x\right)=y\right]\\ x=y^2-16\left[\text{Interchange }x\text{ and }y\right]\\ y^2=x+16\left[\text{Solve for }y\right]\\ y=\pm \sqrt{x+16}\\ f^{-1}\left(x\right)=\pm \sqrt{x+16}\left[\text{Replace }y\text{ with }{f}^{-1}\left(x\right)\right]\end{array}$

Now make a table of values for $f^{-1}\left(x\right)$ as shown below.

$f^{-1}\left(x\right)$	$x$	$-16$	$-10$	$-5$	0	5	10	16
	$y$	0	$\pm 2.45$	$\pm 3.32$	$\pm 4$	$\pm 4.58$	$\pm 5.10$	$\pm 5.66$

Now plot the points and join them with smooth curves to get the graph of $f^{-1}\left(x\right)$ as shown below.