Evaluating a Quadratic Function at a Point

Let the function $f(x)$ be defined as follows:

$$f(x) = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants such that $a \neq 0$, and the function is known to have a minimum point at $x = 2$. Additionally, it is given that $f(2) = 3$.

Let another function $g(x)$ be defined by:

$$g(x) = 2f(x) + 5$$

What is the value of $g(2)$?