Find $f(x)$ if for every $x$: $f(x) + f(\frac {2x-3}{x-1}) = x$

Question

I want to find $f(x)$ if for every $x$ (except one and two):

$$f(x) + f\left(\frac {2x-3}{x-1}\right) = x$$

I know that the answer goes something like $g(x)= \frac {2x-3}{x-1} $ and in conclusion $g(g(g(x)))=x$ But i don't know what to do from there on...

Answer 1

Since $g(g(g(x))) = x$, we have $$\begin{array}{crcll} (1) & f(x) &+& f(g(x)) &= x\\ (2) & f(g(x)) &+& f(g(g(x)) &= g(x)\\ (3) & f(g(g(x)) &+& f(x) &= g(g(x)) \end{array} \quad\xrightarrow{(1)-(2)+(3)}\quad 2 f(x) = x - g(x) + g(g(x)) $$ This leads to $\displaystyle\;f(x) = \frac{x^3-4x^2+5x-3}{2x^2-6x+4}$