9. Problem: Suppose $f(x)$ is a rational function such that $3 f\left(\frac{1}{x}\right)+\frac{2 f(x)}{x}=x^{2}$ for $x \neq 0$. Find $f(-2)$.
Solution: Let $x=\frac{-1}{2}$. Then
$$
\begin{aligned}
3 f(-2)+\frac{2 f\left(\frac{-1}{2}\right)}{\frac{-1}{2}} & =\frac{1}{4} \\
\Rightarrow 3 f(-2)-4 f\left(\frac{-1}{2}\right) & =\frac{1}{4}
\end{aligned}
$$

Let $x=-2$. Then
$$
\begin{array}{l}
3 f\left(\frac{-1}{2}\right)+\frac{2 f(-2)}{-2}=4 \\
\Rightarrow 3 f\left(\frac{-1}{2}\right)-f(-2)=4
\end{array}
$$

Solving this system of equations $\{(1),(2)\}$ for $f(-2)$ yields $f(-2)=\frac{67}{20}$.