Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=1-x^3$
$g(x)=\sqrt[3]{1-x}$
a) Verify that the two functions are inverse functions.
Determine $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f(\sqrt[3]{1-x})=1-(\sqrt[3]{1-x})^3=1-(1-x)=1-1+x=x$
$(g\circ f)(x)=g(f(x))=g(1-x^3)=\sqrt[3]{1-(1-x^3)}=\sqrt[3]{x^3}=x$
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 $y=x$.
The graphs of $f$ and $g$ are symmetric with respect to the line $y=x$; therefore $f$ and $g$ are inverse functions.
