Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=x^3+5$
$g(x)=\sqrt[3]{x-5}$
Compute $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f\left(\sqrt[3]{x-5}\right)=(\sqrt[3]{x-5})^3+5=x-5+5=x$
$(g\circ f)(x)=g(f(x))=g\left(x^3+5\right)=\sqrt[3]{x^3+5-5}=\sqrt[3]{x^3}=x$
We got:
$(f\circ g)(x)=(g\circ f)(x)=x$,
therefore $f$ and $g$ are inverse functions.
