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Let {{mvar|''X''}} be a set and {{math|''f'': ''X'' → ''X''}} be a [[function (mathematics)|function]].
Defining {{math| ''f'' <sup>''n''</sup>}} as the ''n''-th iterate of {{mvar|''f''}} (a notation introduced by [[Hans Heinrich Bürmann]] and [[John Frederick William Herschel]]<!-- in 1813 --><ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Peano_1903"/><ref name="Cajori_1929"/>), where ''n'' is a non-negative integer, by:
<math display="block">f^0 ~ \stackrel{\mathrm{def}}{=} ~ \operatorname{id}_X</math>
and
<math display="block">f^{n+1} ~ \stackrel{\mathrm{def}}{=} ~ f \circ f^{n},</math>
where {{math|id<sub>''X''</sub>}} is the [[identity function]] on {{mvar|''X''}} and {{math|(''f'' {{text| {{math| <math>\circ</math> }} }} ''g'')(''x'') {{=}} ''f'' (''g''(''x''))}} denotes [[function composition]].
Because the notation {{math|''f'' <sup>''n''</sup>}} may refer to both iteration (composition) of the function {{mvar|''f''}} or [[Exponentiation#Iterated functions|exponentiation of the function]] {{mvar|''f''}} (the latter is commonly used in [[trigonometric functions|trigonometry]]), some mathematicians{{citation needed|date=August 2020|reason=Origin? Example authors?}} choose to use {{math|∘}} to denote the compositional meaning, writing {{math|''f''{{i sup|∘''n''}}(''x'')}} for the {{mvar|n}}-th iterate of the function {{math|''f''(''x'')}}, as in, for example, {{math|''f''{{i sup|∘3}}(''x'')}} meaning {{math|''f''(''f''(''f''(''x'')))}}. For the same purpose, {{math|''f'' <sup>[''n'']</sup>(''x'')}} was used by [[Benjamin Peirce]]<ref name="Peirce_1852"/><ref name="Cajori_1929"/><ref group="nb">while {{math|''f'' <sup>(''n'')</sup>}} is taken for the [[Derivative#Lagrange's notation|{{math|''n''}}th derivative]]</ref> whereas [[Alfred Pringsheim]] and [[Jules Molk]] suggested {{math|{{i sup|''n''}}''f''(''x'')}} instead.<ref name="Pringsheim-Molk_1907"/><ref name="Cajori_1929"/><ref group="nb" name="NB_Rucker"/>
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