If $f(x)=2x^3+4$, find $f^{-1}(58)$.
The value $x=f^{-1}(58)$ is the solution to $f(x)=58$.  This means \[2x^3+4=58.\]Subtracting 4 gives \[2x^3=54.\]If we divide by 2 we get  \[x^3=27,\]and the only value that solves this equation is  \[x=\boxed{3}.\]