What is "cyclic function"

In mathematics, a cyclic function ƒ is a function that when iterated some finite number of times yields the identity function, thus:

f(f(f(⋯(f(((⋯(x)⋯) = x 

One can express this as

$f^n(x) = (\,\underbrace{f\circ \cdots \circ f}_{n \text{ iterations}}\,)(x) = x.$

for all values of x in the domain of ƒ. The number of iterations needed is the order of cyclicity, so that if n iterations are needed then one says that ƒ is cyclic of order n.

Cyclic functions can be used in solving problems by substituting a function for its cyclic pair.