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Determine all functions $f: \mathbb{R \setminus\{0,1\}} \rightarrow \mathbb{R} $ satisfying $f(x) + f(\frac{1}{1−x}) = \frac{2(1−2x)}{x(1−x)}$

I have solved this question by doing the substitution $x \rightarrow \frac{1}{1-x}$ twice. I get $f(x) = \frac{x+1}{x-1}$

Now, my question is how to determine all the functions that may be possible or to prove that the function I got is the unique solution to the functional equation. I am also not given any constraints on the function except its range and domain. So I feel like there might be more functions out there but I have no idea how to approach this problem. Any ideas using calculus are also appreciated.

Mohsen Shahriari
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Raahil Rai
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1 Answers1

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Let $f(x)=\frac{x+1}{x-1}$, let $g$ be another function satisfying the equation, and let $h:=f-g$. (that is, $h(x)=f(x)-g(x)$.) Substracting $f(x) + f(\frac{1}{1−x}) = \frac{2(1−2x)}{x(1−x)}$ from $g(x) + g(\frac{1}{1−x}) = \frac{2(1−2x)}{x(1−x)}$, we get that $h(x)+h(\frac1{1-x})=0.$

From here, playing around with the equation, you can conclude that $h(x)=0$, i.e., that $f$ is the unique solution.

Kenta S
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