Answer
See explanations.
Work Step by Step
Step 1. Given $f(x)=x^3+7, g(x)=\sqrt[3] {x-7}$, we have $(f\circ g)(x)=f(g(x))=(\sqrt[3] {x-7})^3+7=x-7+7=x$
Step 2. $(g\circ f)(x)=g(f(x))=\sqrt[3] {x^3+7-7}=x$
Step 3. We conclude that $(f\circ g)(x)= (g\circ f)(x)=x$ for this case.