Chapter 1, Problem 17CT

To determine

To state : whether the function has an inverse function.

Answer to Problem 17CT

Yes, function has an inverse function and the inverse function is $g\left(x\right)=\sqrt[3]{x-8}$.

Explanation of Solution

Given information :

The function is $f\left(x\right)=x^3+8$.

from the above function it can be observed that the function has an inverse function because for every value of $y$ there is an unique x -value.

For inverse function,

  $\begin{array}{l}y=x^3+8\\ y-8=x^3\text{ [subtract 8 both side]}\\ x=\sqrt[3]{y-8}\text{ [root]}\\ g\left(x\right)=\sqrt[3]{x-8}\end{array}$

The inverse function is $g\left(x\right)=\sqrt[3]{x-8}$.