Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=\dfrac{x^3}{4}$
$g(x)=\sqrt[3]{4x}$
Compute $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f\left(\sqrt[3]{4x}\right)=\dfrac{(\sqrt[3]{4x})^3}{4}=\dfrac{4x}{4}=x$
$(g\circ f)(x)=g(f(x))=g\left(\dfrac{x^3}{4}\right)=\sqrt[3]{4\left(\dfrac{x^3}{4}\right)}=\sqrt[3]{x^3}=x$
We got:
$(f\circ g)(x)=(g\circ f)(x)=x$,
therefore $f$ and $g$ are inverse functions.
