R denotes the set of real numbers. Does there exist functions F: R → R such that:
F(F(x)) = x^{2}  2?
If so, find all such functions.
If not, prove that no such function can exist.
Source: American Mathematical Monthly
(In reply to
re: Almost there? by Steve Herman)
Sorry if I skipped some steps.
1) Suppose f(2)=a, then f(f(2))=f(a)=2
Since f(f(a))=a^22
we have a^22=a
solutions a=2 or a=1
So either f(2)=2 or f(2)=1
2) f(2)=2 because f(f(f(2)))=f(f(x)) which implies x^22=2
so x=2 (case 1)
or x=2 (case 2) ***I may have made an error here***

Posted by Jer
on 20151014 10:07:48 