Given the function $f(x)=3x^3+2$, find the value of $x$ so that $f^{-1}(x)=4$.
The equation $f^{-1}(x)=4$ is equivalent to $x=f(4)$.  Therefore we want to find the value $f(4)$. We compute $f(4) = 3 \cdot 4^3 + 2 = \boxed{194}$.