Answer
a) $x^2$; domain $(-\infty,\infty)$
b) $x^2$; domain: $(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=x^3$
$g(x)=x^{2/3}$
Determine the domains $D_f$ and $D_g$ of the two functions:
$D_f=(-\infty,\infty)$
$D_g=(-\infty,\infty)$
a) Find $f\circ g$ and its domain $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(x^{2/3}\right)=(x^{2/3})^3=x^2$
$D_{f\circ g}=(-\infty,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(x^3)=(x^3)^{2/3}=x^2$
$D_{g\circ f}=(-\infty,\infty)$
