Answer
$f^{-1}(x)=\sqrt[3]{x}-1$
Work Step by Step
$f(x)=(x+1)^{3}$
Substitute $f(x)$ by $y$:
$y=(x+1)^{3}$
Solve for $x$:
$\sqrt[3]{y}=x+1$
$x+1=\sqrt[3]{y}$
$x=\sqrt[3]{y}-1$
Interchange $x$ and $y$:
$y=\sqrt[3]{x}-1$
Substitute $y$ by $f^{-1}(x)$:
$f^{-1}(x)=\sqrt[3]{x}-1$