Kezer 08.09.2016 00:02 Find all functions $f$ that is defined on all reals but $\tfrac13$ and $- \tfrac13$ and satisfies \[ f \left(\frac{x+1}{1-3x} \right) + f(x) = x \]for all $x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}$.
weirdo37 08.09.2016 00:33 Setting $\frac{x+1}{1-3x}=:g(x)$ we can show that g(g(g(x)))=x holds. rest is easy (set up a system of equations.)
ZETA_in_olympiad 16.10.2022 14:04 Let $P(x)$ denote $f(\frac{x+1}{1-3x})+f(x)=x.$ $P(\frac{x+1}{1-3x})\implies f(\frac{x-1}{1+3x})+f(\frac{x+1}{1-3x})=\frac{x+1}{1-3x}$ $P(\frac{x-1}{1+3x})\implies f(x)+f(\frac{x-1}{1+3x})=\frac{x-1}{1+3x}$ So solving we get, $f(x)=\frac{9 x^3 + 6 x^2 - x + 2}{18 x^2 - 2}$, which works.