Answer
$f^{-1}(x)=x^3-1$
Work Step by Step
The given function, $
f(x)=\sqrt[3]{x+1}
,$ is equivalent to
\begin{array}{l}\require{cancel}
y=\sqrt[3]{x+1}
.\end{array}
Interchanging the $x$ and $y$ variables and then solving for $y$ result to
\begin{array}{l}\require{cancel}
x=\sqrt[3]{y+1}
\\\\
(x)^3=(\sqrt[3]{y+1})^3
\\\\
x^3=y+1
\\\\
x^3-1=y
\\\\
y=x^3-1
.\end{array}
Hence, the inverse is $
f^{-1}(x)=x^3-1
.$
In the graph above, the black graph is the graph of $f(x)$ and the red graph is the graph of $f^{-1}(x).$
