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