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f (f (...( f (y)) ...)) = a where f (x) = x - 1/x

Source: 10th QEDMO p9 Seniors (8-10. 12. 2011) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 16, 2021
algebrafunctional equationfunctional

Problem Statement

Let X=Q−{−1,0,1}X = Q-\{-1,0,1\}. We consider the function f:X→Xf: X\to X given by f(x)=x−1x.f (x) = x -\frac{1}{x} . Is there an a∈Xa \in X such that for every natural number n there is a y∈Xy \in X with f(f(...(f(y))...))=af (f (...( f (y)) ...)) = a where ff occurs exactly nn times on the left side?
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