A little bit of knowledge...

...can be very frustrating when you do not know what to do with it.

Functions of the form c - x and c/x have the property that they are
self-inverses, that is
f^2(x)=x. It is possible to generate other such functions by taking
two of them, f and g, and generating fgf. It seems natural to wonder
about what is generated by simply taking fg. In particular, working
with 1/x and 1 - x gives f(x)=1 - 1/x and f(x) = 1/(1-x). These
functions are just vertical and horzontal shifts of f(x)= -1/x.
However, they have the curious property that they satisfy f^3(x)=x. Is
there any significance to this? Is there a branch of mathematics that
studies such things? Can functions be found that satisfy f^n(x)=x for
arbitrary n?

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