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I have come across a problem which asks to find $f(x)$ such that $f(f(x))=-x$. Nothing I can find has anything pertaining to the composition of two identical functions. Is there a way that I can dissect this in order to help in finding a possible $f(x)$?

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Here are some possible answers:

  • Multiply by sqrt(-1).

  • Rotate by 90 degrees.

  • Map an even number x to x+1, and an odd number to 1-x.

I don't think it is possible if f is continuous on real numbers.

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I would the Fourier Transform help? If I recall correctly (note that I am 15), the Fourier Transform is just converting a function from a time to a frequency domain. Where would this be applicable to my situation? – fr00ty_l00ps Nov 8 '13 at 14:26
You are right and I am wrong. I will edit my answer. – apt1002 Nov 9 '13 at 3:10
Another question, how would the last two be represented and/or transformed into a formula or equation? – fr00ty_l00ps Nov 11 '13 at 14:25
You could represent the rotation as a matrix. The even/odd one is best left as a list of cases, I reckon. – apt1002 Nov 13 '13 at 18:44
up vote 0 down vote accepted

Through late night serendipity and later verification, I have found that $f(x)=ix$ works for all real numbers.

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