Let's call a real-valued continuous function an n-th identity root when it generates the identity function after exactly n compositions with itself. For example f(x)=-x is a 2nd identity root because f(f(x))=x.

1. The function f(x)=1-1/x is a 3rd identity root. Unfortunately it is undefined at x=0. Are there identity roots for n>2 which are defined for all real numbers?

2. For a given real number c and n>1, give an example of an n-th identity root which is defined for all real numbers except c. How many such roots exist?

Congratulations to Joel, impressive solution! I am especially pleased that Joel's example is quite different from mine, which means I could learn something new about my own puzzle.

If anyone still cares for this problem: You might be interested to hear that the second part has an alternative solution which can be expressed as a single, "nice" formula, i.e. without piecewise definition. To find it, think about possible n-th identity roots defined on a circle (rather than the line of real numbers). Then "lift" this function over to the line of real numbers.

OK, no-one really cares anyway, right...

Posted by JLo on 2006-11-08 17:43:45