Answer
$f^{-1}(x)=(x-2)^{3}$
Work Step by Step
$f(x)=2+\sqrt[3]{x}$
Rewrite this expression as $y=2+\sqrt[3]{x}$ and solve for $x$:
$y=2+\sqrt[3]{x}$
Take $2$ to the left side:
$y-2=\sqrt[3]{x}$
$\sqrt[3]{x}=y-2$
Cube both sides:
$(\sqrt[3]{x})^{3}=(y-2)^{3}$
$x=(y-2)^{3}$
Interchange $x$ and $y$:
$y=(x-2)^{3}$
The inverse of the original function is $f^{-1}(x)=(x-2)^{3}$