Chapter 1.6, Problem 76E

To determine

To find : the inverse function.

Answer to Problem 76E

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

Explanation of Solution

Given information :

The function $f\left(x\right)=x^3+1$.

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

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

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

Now by using graphing utility the graph of function $f\text{ and }{f}^{-1}$ can be obtained as:

  

Form the above graph it can be observed that the graph of function $f\text{ and }g$ are the reflection of each other in the line $y=x$.