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A Rational Function that Iterates to the Identity

​
x
1
curve parameter a
2
zoom
6
The function
f
a
(x)=
ax-(1+a+
2
a
)
x-(1+a)
has the property that
f(f(f(x)))=x
for all
x
in the extended reals (i.e.,
x
can be infinite as well as finite). In the graph,
x
can be taken to be any one of the three points on the
x
axis. The sides of the polygon trace out the steps of the triple composition.
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