Answer
a) $x^{5/4}$; domain $[0,\infty)$
b) $x^{5/4}$; domain: $[0,\infty)$
Work Step by Step
We are given the functions:
$f(x)=x^5$
$g(x)=\sqrt [4]x=x^{1/4}$
Determine the domains $D_f$ and $D_g$ of the two functions:
$D_f=(-\infty,\infty)$
$D_g=[0,\infty)$
a) Find $f\circ g$ and its domain $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(x^{1/4}\right)^5=x^{5/4}$
$D_{f\circ g}=[0,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(x^5)=(x^5)^{1/4}=x^{5/4}$
$D_{g\circ f}=[0,\infty)$
