Answer
$f[g(x)]=g[f(x)]=x$. This means that $f(x)$ and $g(x)$ are inverses of each other.
Both $f(x)$ and $g(x)$ are continuous on the set of real numbers, so there are no restrictions to their domains.
Work Step by Step
We wish to plug $f(x)$ into $g(x)$ to obtain:
$$\displaystyle g[f(x)]=g(x^3−8)\\ =\sqrt[3] {x(x^3−8)+8}-3\\=\sqrt[3] {x^3} \\=x$$
We wish to plug $g(x)$ into $f(x)$ to obtain:
$$f[(g(x)]=f (\sqrt [3]{x+8} ) \\ =(\sqrt[3] {x+8)^3}-8 \\=x+8-8 \\=x$$
We see that $f[g(x)]=g[f(x)]=x$. This means that $f(x)$ and $g(x)$ are inverses of each other.
Both $f(x)$ and $g(x)$ are continuous on the set of real numbers, so there are no restrictions to their domains.
