Answer
$f(g(x))=x$ and $g(f(x))=x$ therefore $f$ 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 domain restrictions.
Work Step by Step
Substitute $g(x)$ to $x$ in $ f(x)$ to obtain:
\begin{align*}
f\left(g(x)\right)&=2\left(\frac{1}{2}x-3\right)+6\\
&=x-6+6\\
&=x
\end{align*}
Substitute $f(x)$ to $x$ in $g(x)$ to obtain:
\begin{align*}
g\left(f(x)\right)&=\frac{1}{2}(2x+6)-3\\
&=x+3-3\\
&=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.