Answer
$
f^{-1}(V)=\sqrt[3]{\frac{3 V}{4 \pi}}
$
Work Step by Step
We solve the equation $V=f(r)=\frac{4}{3} \pi r^3$ for $r$. Divide both sides by $\frac{4}{3} \pi$ and then take the cube root to get
$$
r=\sqrt[3]{\frac{3 V}{4 \pi}}
$$
So
$$
f^{-1}(V)=\sqrt[3]{\frac{3 V}{4 \pi}}
$$
