tommy1729 Wrote:f(x) =/= x
f(f(x)) = x
exp(f(x)) = f(exp(x))
give examples of Coo f(x) satisfying all the above 3 equations at once.
It may be interesting for you that there is no continuous solution of \( f(f(f(x)))=x \).
And that each solution of \( f(f(x))=x \) is strictly decreasing with a fixed point.
You can read further on this subject (keyword Babbage equation, involution) in
Kuczma, Iterative functional equations.