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.
$\endgroup$1 Answer
$\begingroup$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.
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