Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=\dfrac{x^3}{3}$
$g(x)=\sqrt[3] {3x}$
a) Verify that the two functions are inverse functions.
Determine $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f\left(\sqrt[3] {3x}\right)=\dfrac{(\sqrt[3] {3x})^3}{3}=\dfrac{3x}{3}=x$
$(g\circ f)(x)=g(f(x))=g\left(\dfrac{x^3}{3}\right)=\sqrt[3]{3\left(\dfrac{x^3}{3}\right)}=\sqrt[3]{x^3}=x$
We got:
$(f\circ g)(x)=(g\circ f)(x)=x$,
therefore the two functions are inverse functions.
b) Graph $f$ and $g$ and the line $y=x$.
As the graphs of $f$ and $g$ are symmetric with respect to the line $y=x$, $f$ and $g$ are inverse functions.
