I've done part (i) pretty easily but I've no idea about (ii). I think I want to use the earlier hint about the generators but I can't seem to get anywhere.
2026-03-27 13:45:38.1774619138
Solving a functional equation using Mobius transformations
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The functional equation with $z = x$ is $$f\left(\dfrac{1}{1-x}\right)+f\left(\dfrac{x-1}{x}\right) = \dfrac{x}{x-1}.$$
The function equation with $z = \dfrac{1}{1-x}$ is $$f\left(\dfrac{x-1}{x}\right)+f\left(x\right) = \dfrac{1}{x}.$$
The function equation with $z = \dfrac{x-1}{x}$ is $$f\left(x\right)+f\left(\dfrac{1}{1-x}\right) = 1-x.$$
Can you solve this system of equations for $f(x)$?