Answer
$f^{-1}(x)=x^3-1$
Work Step by Step
Replace $f(x)$ by $y$ to have:
$y=\sqrt[3]{x+1}$
Interchange $x$ and $y$ to have:
$x = \sqrt[3]{y+1}$
Raise both sides to the 3rd power to have:
$x^3=y+1
\\x^3-1=y
\\y=x^3-1$
Replace $y$ with $f^{-1}(x)$ to have:
$f^{-1}(x)=x^3-1$
