# A Rational Function that Iterates to the Identity

A Rational Function that Iterates to the Identity

The function has the property that for all in the extended reals (i.e., can be infinite as well as finite). In the graph, can be taken to be any one of the three points on the axis. The sides of the polygon trace out the steps of the triple composition.

f(x)=

a

ax-(1+a+a)

2

x-(1+a)

f(f(f(x)))=x

x

x

x

x