6. For all real numbers $x$ satisfying $|x| \neq 1$, the function $f(x)$ satisfies
$$
f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x .
$$

Find all possible $f(x)$.
(12th Korean Mathematical Olympiad)
Solve: Let $x$ be $\frac{x-3}{x+1}$ and substitute into the original equation, we get
$$
f\left(\frac{3+x}{1-f}\right)+f(x)=\frac{x-3}{x+1} \text {. }
$$

Let $x$ be $\frac{3+x}{1-x}$ and substitute into the original equation, we have
$$
f(x)+f\left(\frac{x-3}{x+1}\right)=\frac{3+x}{1-x} \text {. }
$$
(1) + (2), and substitute the original equation, we get
$$
f(x)=\frac{x^{3}+7 x}{2-2 x^{2}},|x| \neq 1 .
$$