Answer
$g^{-1}(x)=\sqrt[3]{x}-1$
Work Step by Step
Let $y=g(x).$ Then the given function, $g(x)=(x+1)^3,$ becomes $y=(x+1)^3.$
To get the inverse, interchange the $x$ and $y$ variables and then solve for $y.$ That is,
\begin{align*}\require{cancel}
x&=(y+1)^3
&(\text{interchange $x$ and $y$})
\\
\sqrt[3]{x}&=\sqrt[3]{(y+1)^3}
&(\text{Solve for $y$})
\\
\sqrt[3]{x}&=y+1
\\
\sqrt[3]{x}-1&=y
\\
y&=\sqrt[3]{x}-1
.\end{align*}
Hence, the inverse, $g^{-1}(x),$ is $g^{-1}(x)=\sqrt[3]{x}-1$.
