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 $y$: $x=y^3+1\\x-1=y^3\\y=f^{-1}(x)=\sqrt[3] {x-1}$.
$f[f^{-1}(x)]=f[\sqrt[3] {x-1}]=(\sqrt[3] {x-1})^3+1=x-1+1=x$. Hence we verified what we had to.