Determine all functions $f$ satisfying the functional relation $f(x)+f\left(\frac{1}{1-x}\right)=\frac{2(1-2x)}{x(1-x)}$

Question

The question is : Determine all functions f satisfying the functional relation $f(x)+f\left(\frac{1}{1-x}\right)=\frac{2(1-2x)}{x(1-x)}$ where $x$ is a real number . Also $x$ is not equal to $0$ or $1$. Here $f:\mathbb{R}-\{0,1\} \to \mathbb{R}$.

I try the following in many ways. But I can't proceed.I put many things at the place of $x$ to get the answer but I failed again and again.

Answer 1

Let $P(x)$ be the assertation of the function equation. Then:

$$P(x) \implies f(x)+f\left(\frac{1}{1-x}\right)=\frac{2(1-2x)}{x(1-x)}$$

$$P\left(\frac{1}{1-x}\right) \implies f\left(\frac{1}{1-x}\right) + f\left(\frac{x-1}{x}\right) = \frac{2(x+1)(1-x)}{x}$$

$$P\left(\frac{x-1}{x}\right) \implies f\left(\frac{x-1}{x}\right) + f(x) = \frac{2x(2-x)}{x-1}$$

Adding the first and third equation and subtracting the second one will yield $2f(x)$ on the LHS and you should be able to manipulate the RHS.