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?

(In reply to There's still something to find... by JLo)

how nice are we talking?

tan(arctan(x-c+tan(pi/2-pi/n))+pi/n)

seems to do the trick, though it seems like it isn't really using a circle and dealing with the c part seems very ugly...

Posted by Joel on 2006-11-08 19:26:41