<s>[INST] Functional equation: $f\left(\frac{x-1}{x}\right)+ f\left(\frac{1}{1-x}\right)= 2- 2x$ There is a function given $f\left(\dfrac{x-1}{x}\right)+ f\left(\dfrac{1}{1-x}\right)= 2- 2x
,f\colon \Bbb R\setminus\{0,1\}\to \Bbb R$ How many fuction exist? I have no idea how to start
 [/INST] $f\left(\dfrac{x-1}{x}\right)+ f\left(\dfrac{1}{1-x}\right)= 2- 2x\tag{1}$
Replace $\displaystyle x\rightarrow \frac{x-1}{x} = 1-\frac{1}{x}$ in $(1)$
$\displaystyle f\left(\frac{1}{1-x}\right)+f\left(x\right) = \frac{2}{x}\tag{2}$
Similarly Replace $\displaystyle x\rightarrow \frac{x-1}{x} = 1-\frac{1}{x}$ in $(2)$
$\displaystyle f(x)+f\left(\frac{x-1}{x}\right) = \frac{2x}{x-1}\tag{3}$
Now $(1)-(2)-(3),$ we get
$$\displaystyle -2f(x) = 2-2x-\frac{2}{x}-\frac{2x}{x-1} = -2\left(\frac{x^3+x-1}{x^2-x}\right)$$
So $\displaystyle f(x) = \frac{x^3+x-1}{x^2-x}$ for all $x\in \mathbb{R}\setminus\{0,1\}$
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