Find all functions of with the property

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- Oct 18th 2008, 03:23 AMalexmahoneFunctional equation
Find all functions of with the property

- Oct 18th 2008, 04:33 AMmr fantastic
- Oct 18th 2008, 06:09 AMalexmahone
- Oct 18th 2008, 12:22 PMShowcase_22

Does this help in any way? I was thinking that somehow I could get the to cancel out but I couldn't get it to work.

The RHS can be multiplied by to give:

but I don't think that's helping.

It's an interesting question. I hope someone here can solve it!

EDIT: Perhaps you could try case analysis. When x=0:

[tex]f(f(y))=(f(0))^2+y[/tex]

f(0) has an unknown value so it might not help. However, we have successfully eliminated a term which is beneficial! - Oct 18th 2008, 01:28 PMmr fantastic
Hmmm ..... you're right.

I know f(x) = x works (an involution, by the way). I haven't checked involutions of the form f(x) = (ax + b)/(cx + d), where there's a relationship between a, b, c, and d that I can't be bothered remembering .... might be worth a check. - Oct 19th 2008, 02:53 PMSimonM
I believe that in the case of f(0)=0 a function being an involution is a necessary but not sufficient criteria..

- Oct 20th 2008, 01:22 AMNonCommAlg