<s>[INST] Find all functions that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$ I found the following task in a book and I would be interested if someone has an idea to solve it:
Find all the functions $f$ that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$.
My ideas:
Assuming that $f$ is a power series or making a substitution.
I tried several approaches, but without success.
 [/INST] Hint:
Consider $T(x+1)=\dfrac{T(x)+4}{1-T(x)}$ ,
Let $T(x)=U(x)+1$ ,
Then $U(x+1)+1=\dfrac{U(x)+5}{-U(x)}$
$U(x+1)+1=-1-\dfrac{5}{U(x)}$
$U(x+1)=-2-\dfrac{5}{U(x)}$
Let $U(x)=\dfrac{V(x+1)}{V(x)}$ ,
Then $\dfrac{V(x+2)}{V(x+1)}=-2-\dfrac{5V(x)}{V(x+1)}$
$\dfrac{V(x+2)}{V(x+1)}=-\dfrac{2V(x+1)+5V(x)}{V(x+1)}$
$V(x+2)+2V(x+1)+5V(x)=0$
$V(x)=\theta_1(x)(-1+2i)^x+\theta_2(x)(-1-2i)^x$ , where $\theta_1(x)$ and $\theta_2(x)$ are arbitrary periodic functions with unit period
$V(x)=\theta_1(x)e^{x\ln(-1+2i)}+\theta_2(x)e^{x\ln(-1-2i)}$ , where $\theta_1(x)$ and $\theta_2(x)$ are arbitrary periodic functions with unit period
$V(x)=\theta_1(x)e^{\frac{x\ln5}{2}+(\pi-\tan^{-1}2)ix}+\theta_2(x)e^{\frac{x\ln5}{2}-(\pi-\tan^{-1}2)ix}$ , where $\theta_1(x)$ and $\theta_2(x)$ are arbitrary periodic functions with unit period
$V(x)=\Theta_1(x)5^\frac{x}{2}\sin((\tan^{-1}2)x)+\Theta_2(x)5^\frac{x}{2}\cos((\tan^{-1}2)x)$ , where $\Theta_1(x)$ and $\Theta_2(x)$ are arbitrary periodic functions with unit period
$\therefore T(x)=\dfrac{\Theta_1(x+1)5^\frac{x+1}{2}\sin((\tan^{-1}2)(x+1))+\Theta_2(x+1)5^\frac{x+1}{2}\cos((\tan^{-1}2)(x+1))}{\Theta_1(x)5^\frac{x}{2}\sin((\tan^{-1}2)x)+\Theta_2(x)5^\frac{x}{2}\cos((\tan^{-1}2)x)}+1$ , where $\Theta_1(x)$ and $\Theta_2(x)$ are arbitrary periodic functions with unit period
$T(x)=\dfrac{\Theta(x)\sqrt5\sin((\tan^{-1}2)(x+1))+\sqrt5\cos((\tan^{-1}2)(x+1))}{\Theta(x)\sin((\tan^{-1}2)x)+\cos((\tan^{-1}2)x)}+1$ , where $\Theta(x)$ is an arbitrary periodic function with unit period
Hence $f(2-2\tan((\tan^{-1}2)(x+1)))+f(2-2\tan((\tan^{-1}2)x))=2-2\tan((\tan^{-1}2)x)$
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