Chapter 5.6, Problem 26E

To determine

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

Answer to Problem 26E

The inverse function is $f^{-1}\left(x\right)=\sqrt[3]{x}-4$ .

Explanation of Solution

Given information:

the function is $f\left(x\right)={\left(x+4\right)}^3$ .

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

  $\begin{array}{l}f\left(x\right)={\left(x+4\right)}^3\\ y={\left(x+4\right)}^3\text{ [substitute }f\left(x\right)=y]\\ x+4=\sqrt[3]{y}\text{ [cube root]}\\ x=\sqrt[3]{y}-4\text{ [subtract and simplify]}\\ f^{-1}\left(x\right)=\sqrt[3]{x}-4\text{ [simplify]}\end{array}$

Thus, the inverse is $f^{-1}\left(x\right)=\sqrt[3]{x}-4$ .

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

$x$	$f\left(x\right)$
$-1$	$27$
$0$	$64$
$-2$	$8$

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: