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:
$$\displaystyle g[f(x)]=\frac{-1}{2} (3-2x-3) \\=\dfrac{-1}{2}(-2x)\\=x$$
We wish to plug $g(x)$ into $f(x)$ to obtain:
$$f[(g(x)]=3-2(\dfrac{-1}{2}(x-3))\\=3+(x-3) \\=x$$
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.
