Solve in English

To find $F(2)$, we substitute $x=2$ into the expression for $F(x)$.

$$F(2) = 2(2) + 3$$ $$F(2) = 4 + 3$$ $$F(2) = 7$$

The value of $F(2)$ is $\boxed{7}$.

To find the inverse function $g^{-1}(x)$, we set $y = g(x)$, then swap $x$ and $y$ and solve for $y$.
Let $y = x-1$.
Swap $x$ and $y$:

$$x = y-1$$

Now, solve for $y$:

$$y = x+1$$

So, the inverse function is $g^{-1}(x) = x+1$.
The inverse function of $g(x)$ is $\boxed{g^{-1}(x) = x+1}$.

The notation $fg(x)$ represents the composition of functions $F(g(x))$. We substitute the expression for $g(x)$ into $F(x)$.

$$fg(x) = F(g(x))$$

We know that $g(x) = x-1$.

$$fg(x) = F(x-1)$$

Now, we substitute $(x-1)$ in place of $x$ in the expression for $F(x) = 2x+3$:

$$fg(x) = 2(x-1) + 3$$ $$fg(x) = 2x - 2 + 3$$ $$fg(x) = 2x + 1$$

The composite function $fg(x)$ is $\boxed{fg(x) = 2x+1}$.