Can you find a continuous function f such that f(f(x)) = x^2 + 1?

Answer 1

A very interesting question. For example if f(x) = x^(sqrt(2)) then f(f(x)) = x^2 however I am not sure what to do about the +1. If f(x) is x^(sqrt(2)) + 1 then I end up trying to raise a polynomial to the sqrt(2) power. I don't know how to do that. I have a funny feeling that this cannot be done, but cannot prove it.

Answer 2

The following information might be useful to prove that there is no continuous function f, such that f(f(x))=x^2+1:

If f(f(x))=x^2+1, that would mean that f(x) and f(x) would have to be inverses of each other. In other words, f(x) would have to be an inverse of itself.

Answer 3

I have to confess that I've been thinking about this one for a while too, and I haven't come up with anything good. I'll keep trying though.

I don't understand TKD Girl's assertion that f(x) has to be its own inverse, though. Nor do I understand why that would prove anything, as I know at least one function that is its own inverse: f(x) = 1/x.

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Edit 3 days later: OK, I just can't think of any way to figure this out. rscanner came closer than anyone else.