Answer
$f^{-1}(x)=\sqrt[3]{x-1}$
Work Step by Step
We are given the function:
$f(x)=x^3+1$
Determine the inverse of $f$:
$y=x^3+1$
Interchange $x$ and $y$:
$x=y^3+1$
$y^3=x-1$
$y=\sqrt[3] {x-1}$
$f^{-1}(x)=\sqrt[3]{x-1}$
Compute $f\circ f^{-1}$ and $f^{-1}\circ f$:
$(f\circ f^{-1})(x)=f(f^{-1}(x))=f(\sqrt[3]{x-1})=(\sqrt[3]{x-1})^3+1=x-1+1=x$
$(f^{-1}\circ f)(x)=f^{-1}\left(x^3+1\right)=\sqrt[3]{x^3+1-1}=\sqrt[3]{x^3}=x$
Thus, the functions are inverses.
