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
I haven't quite proven that no such function is possible.
f(f(x))=x for only x=2 or x=1
which means
[1] f(2)=2 and f(1)=1
or
[2] f(2)=1 and f(1)=2
In case [1] f(2)=2
[since if f(2)=a besides 2 then f(a)=2 and f(f(a))=2 which would force a=2, a contradiction.]
but them what about f(0)? Let f(0)=b, but then f(b)=2 so that f(f(b))=2 so b is either 2 or 2. That would force f(0)=2, a contradiction since f(f(0))=2 not 2.
In case [2] if f(2)=1 we get the same f(0) contradiction as above
if f(2) is not 1 I have not yet found a contradiction.
Edited on October 13, 2015, 2:02 pm

Posted by Jer
on 20151013 14:01:28 