Answer
See below.
Work Step by Step
If we want to get a formula for the inverse function, we swap $x$ and $y$ and solve for $y$.
Hence here after swapping $x$ and $y$: $x=y^3-8\\x+8=y^3\\y=f^{-1}(x)=\sqrt[3] {x+8}$.
$f[f^{-1}(x)]=f[\sqrt[3] {x+8}]=(\sqrt[3] {x+8})^3-8=x+8-8=x$. Hence we verified what we had to.