I noticed that for the (real) function f ( x ) = 1 1 − x , f ( f ( f ( x ) = f 3 ( x ) = x for all real x.This surprised me, and I was naturally curious about rational functions that elicit the identity after 4 or 5 or, in general, n iterations.I looked at rational functions of the form f ( x ) = 1 z − x for z ∈ R and wondered for which z did f n ( x ) := f ( f ( . . . f ( f ( x ) ) . . . ) ) = x.After playing around for a bit on Wolfram and Desmos, I came up with the following conjecture: If n ∈ N and n ≥ 2 and f ( x ) = 1 2 cos ⁡ ( π n ) − x , then f n ( x ) = x.Note that I do not claim that f is the only rational function of the form f ( x ) = 1 z − x where z ∈ R .So I have a few questions:1) Is this conjecture true?2) How would someone go about proving it, if it is true?3) Is this a well-known theorem or a somewhat-immediate corollary or special case of a well-known theorem? If so, what is that theorem?4) How would I find all z ∈ R such that if f ( x ) = 1 z − x ,then f n ( x ) = x? Is such a thing easy to do?5) What field of math would ask questions like this?

Question

I noticed that for the (real) function   f  (  x  )  =      1          1      −      x      ,   f  (  f  (  f  (  x  )  =      f    3    (  x  )  =  x for all real   x.This surprised me, and I was naturally curious about rational functions that elicit the identity after 4 or 5 or, in general,   n iterations.I looked at rational functions of the form   f  (  x  )  =      1          z      −      x       for   z  ∈      R   and wondered for which   z did       f    n    (  x  )  :=  f  (  f  (  .  .  .  f  (  f  (  x  )  )  .  .  .  )  )  =  x.After playing around for a bit on Wolfram and Desmos, I came up with the following conjecture: If   n  ∈      N   and   n  ≥  2 and   f  (  x  )  =      1          2      cos      ⁡      (              π        n            )      −      x      , then       f    n    (  x  )  =  x.Note that I do not claim that   f is the only rational function of the form   f  (  x  )  =      1          z      −      x       where   z  ∈      R  .So I have a few questions:1) Is this conjecture true?2) How would someone go about proving it, if it is true?3) Is this a well-known theorem or a somewhat-immediate corollary or special case of a well-known theorem? If so, what is that theorem?4) How would I find all   z  ∈      R   such that if   f  (  x  )  =      1          z      −      x      ,then       f    n    (  x  )  =  x? Is such a thing easy to do?5) What field of math would ask questions like this?

Alessandra Weeks · Accepted Answer

The conjecture seems to be true and if it is, it could be proved by mathematical induction.I checked it for up to   n  =  4 and it worked. There are other values for   z for example for   n  =  3, we also have   z  =  −  1 and for   n  =  4 we have   z  =  ±      2   and   z  =  0Note that with       f    3    =  f we have also       f    5    =  f and       f    7    =  f,...You find these type of questions in discrete dynamical systems.

Quintacj · Accepted Answer

Thanks

I noticed that for the (real) function f ( x ) = 1 1 −<!--

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