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Function Existence Exercise (Posted on 2015-10-11) Difficulty: 3 of 5
R denotes the set of real numbers. Does there exist functions F: R → R such that:
F(F(x)) = x2 - 2?
If so, find all such functions.
If not, prove that no such function can exist.

Source: American Mathematical Monthly

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Almost there? | Comment 1 of 4
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 2015-10-13 14:01:28

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