Chapter 5.6, Problem 28E

To determine

Tofind: the inverse of the function and graph the function and inverse function.

Answer to Problem 28E

The inverse function is $f^{-1}\left(x\right)=\sqrt[6]{-y}$ .

Explanation of Solution

Given information:

the function is $f\left(x\right)=-x^6$ .

Calculation: to find the inverse function, first substitute $y=f\left(x\right)$ then simplify,

  $\begin{array}{l}f\left(x\right)=-x^6\\ y=-x^6\text{ [substitute }f\left(x\right)=y]\\ x^6=-y\text{ [dmultiply by }-1\text{]}\\ x=\sqrt[6]{-y}\text{ [simplify]}\\ f^{-1}\left(x\right)=\sqrt[6]{-y}\text{ [simplify]}\end{array}$

Thus, the inverse is $f^{-1}\left(x\right)=\sqrt[6]{-y}$ .

To draw the graph of the function and inverse function, first use the plotting point method,

$x$	$f\left(x\right)$
$1$	$-1$
$0$	$0$
$2$	$-64$

Now by jointing these points, the graph of function can be obtained and the graph of the inverse function is the reflection in the line $y=x$ .

Thus, the graph of function and inverse function can be observed as: