Answer
$a)$ $f^{-1}(x)=x^{3}$
$b)$ $f(f^{-1}(x))=f^{-1}(f(x))=x$
Work Step by Step
$f(x)=\sqrt[3]{x}$
$a)$
Substitute $f(x)$ by $y$:
$y=\sqrt[3]{x}$
Interchange $x$ and $y$:
$x=\sqrt[3]{y}$
Solve for $y$:
$x^{3}=(\sqrt[3]{y})^{3}$
$x^{3}=y$
$y=x^{3}$
Substitute $y$ by $f^{-1}(x)$:
$f^{-1}(x)=x^{3}$
$b)$
Verify the equation for the inverse function found by finding $f(f^{-1}(x))$. Substitute $x$ by $f^{1}(x)$ in $f(x)$ and simplify:
$f(f^{-1}(x))=\sqrt[3]{x^{3}}=x$
Complete the verification process by finding $f^{-1}(f(x))$. Substitute $x$ by $f(x)$ in $f^{-1}(x)$ and simplify:
$f^{-1}(f(x))=(\sqrt[3]{x})^{3}=x$
Since $f(f^{-1}(x))=f^{-1}(f(x))=x$, the equation for the inverse function found is correct.