HMMT February

Let $x = \frac{-1}{2}$. Then $$\begin{align*} 3f(-2) + \frac{2f\left(\frac{-1}{2}\right)}{\frac{-1}{2}} = & \frac{1}{4} \\ \Rightarrow 3f(-2) - 4f\left(\frac{-1}{2}\right) & = \frac{1}{4} \tag{1} \end{align*}$$ Let $x = -2$. Then $$\begin{align*} & 3f\left(\frac{-1}{2}\right) + \frac{2f(-2)}{-2} = 4 \\ \Rightarrow 3f\left(\frac{-1}{2}\right) - f(-2) & = 4 \tag{2} \end{align*}$$ Solving this system of equations $\{(1),(2)\}$ for $f(-2)$ yields $f(-2) = \frac{67}{20}$.