We start with the equation for the composition of functions:
\[
\frac{1}{2}g(f(x)) = 2x^2 - 5x + 2
\]
This implies:
\[
g(f(x)) = 4x^2 - 10x + 4
\]
Assuming \( f(x) \) is quadratic, we get:
\[
(f(x))^2 + f(x) - (4x^2 - 10x + 6) = 0
\]
Solving this quadratic equation for \( f(x) \):
\[
f(x) = \frac{-1 \pm \sqrt{1 + 4(4x^2 - 10x + 6)}}{2}
\]
\[
f(x) = \frac{-1 \pm \sqrt{16x^2 - 40x + 25}}{2}
\]
\[
f(x) = \frac{-1 \pm (4x - 5)}{2}
\]
Breaking this into two cases, we take the positive root:
\[
f(x) = \frac{-1 + 4x - 5}{2} = 2x - 3
\]
Thus, we find that:
\[
f(x) = 2x - 3
\]
Thus, the correct answer is Option A.