Cyclic function

Further information: functional notation

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

$f(f(f(\cdots (f(((\cdots (x)\cdots )=x\,$

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 f. The number of iterations needed is the order of cyclicity, so that if n iterations are needed then one says that f is cyclic of order n. A cyclic function of order 2 is called an involution.

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

References

• Cyclic Functions at AoPS
• Merriam Webster
• Ruscyzk, Robert (2010). Intermediate Algebra. AoPS Incorporated. Search this book on

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📰 Article(s) of the same category(ies)

• Integer
• Natural number

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