Answer
a) $\sqrt[3]{x^3-4}$; domain $(-\infty,\infty)$
b) $x-4$; domain: $(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=\sqrt[3]{x-5}$
$g(x)=x^3+1$
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 $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(x^3+1\right)=\sqrt[3]{x^3+1-5}=\sqrt[3]{x^3-4}$
$D_{f\circ g}=(-\infty,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(\sqrt[3]{x-5})=(\sqrt[3]{x-5})^3+1=x-5+1=x-4$
$D_{g\circ f}=(-\infty,\infty)$
