Question

Find the inverse function of $f\left(x\right)=x^3+1$

Short answer

$f^{-1}\left(x\right)=\sqrt[3]{x-1}$

Step-by-step solution

Step: 1

Given$f\left(x\right)=x^3+1$

Since it is a one-to-one function, inverse exists.

Step: 2

$\begin{array}{r}\mathrm{We}\mathrm{define}\mathrm{a}=\sqrt[3]{\mathrm{b}-1}\\ \mathrm{f}\left(\mathrm{a}\right)=(\sqrt[3]{\mathrm{b}-1}{)}^3+1\\ =\mathrm{b}-1+1\\ =\mathrm{b}\end{array}$

It is onto.

Step: 3

Let

$\begin{array}{r}y=x^3+1\\ \Rightarrow y^3+1=x\\ \Rightarrow y^3=x-1\\ \Rightarrow y=\sqrt[3]{x-1}\\ f^{-1}\left(x\right)=\sqrt[3]{x-1}\end{array}$