Answer
$f(g(x))=x$ and $g(f(x))=x$ so $f$ and $g$ are inverses of each other.
Since both functions have the set of real numbers as a domain, then their inverses have no domain restrictions.
Work Step by Step
Substitute $g(x)$ to $x$ in $ f(x)$ to obtain:
\begin{align*}
f\left(g(x)\right)&=3-2\left(-\frac{1}{2}(x-3)\right)\\
&=3+\frac{2}{2}(x-3)\\
&=3+(x-3)\\
&=x
\end{align*}
Substitute $f(x)$ to $x$ in $g(x)$ to obtain:
\begin{align*}
g\left(f(x)\right)&=-\frac{1}{2}\left(3-2x-3\right)\\
&=-\frac{1}{2}(-2x)\\
&=x
\end{align*}
Since $f(g(x))=g(f(x))=x$, then $f$ and $g$ are inverses of each other.
Both $f(x)$ and $g(x)$ have the set of real numbers as a domain so the inverse has no restrictions.