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Thus one obtains a function ''g'' o ''f'': ''X'' → ''Z'' defined by (''g'' o ''f'')(''x'') = ''g''(''f''(''x'')) for all ''x'' in ''X''. The notation ''g'' o ''f'' is read as "''g'' circle ''f''" or "''g'' composed with ''f''". The composition of functions is always Associative . That is, if ''f'', ''g'', and ''h'' are three functions with suitably chosen domains and codomains, then ''f'' o (''g'' o ''h'') = (''f'' o ''g'') o ''h''. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. As a result the set of with respect to the composition operator. The functions ''g'' and ''f'' Commute with each other if ''g'' o ''f'' = ''f'' o ''g''. In general, composition of functions will not be commutative. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, only when ; for all negative , the first expression is undefined. (But Inverse Function s always commute to produce the Identity Mapping .) Derivative s of compositions involving differentiable functions can be found using the Chain Rule . "Higher" derivatives of such functions are given by Faà Di Bruno's Formula . EXAMPLE As an example, suppose that an airplane's height at time ''t'' is given by the function ''h''(''t'') and that the oxygen concentration at height ''x'' is given by the function ''c''(''x''). Then (''c'' o ''h'')(''t'') describes the oxygen concentration around the plane at time ''t''. FUNCTIONAL POWERS If ''Y''⊂''X'' then ''f'' may compose with itself; this is sometimes denoted ''f'' 2. Thus: :(''f'' o ''f'')(''x'') = ''f''(''f''(''x'')) = ''f'' 2(''x'') :(''f'' o ''f'' o ''f'')(''x'') = ''f''(''f''(''f''(''x''))) = ''f'' 3(''x'') Repeated composition of a function with itself is sometimes called function Iteration . The functional Powers ''f'' o ''f'' ''n'' = ''f'' ''n'' o ''f'' = ''f'' ''n''+1 for Natural ''n'' follow immediately.
Note: If ''f'' takes its values in a Ring (in particular for real or complex-valued ''f'' ), there is a risk of confusion, as ''f n'' could also stand for the ''n''-fold product of ''f'', e.g. ''f'' 2(''x'') = ''f''(''x'') · ''f''(''x''). (For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in Trigonometry , this superscript notation represents standard Exponentiation when used with Trigonometric Function s: sin2(''x'') = sin(''x'') · sin(''x''). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan (≠ 1/tan). In some cases, an expression for ''f'' in ''g''(''x'') = ''f'' ''r''(''x'') can be derived from the rule for ''g'' given non-integer values of ''r''. This is called Fractional Iteration . Iterated Function s occur naturally in the study of Fractals and Dynamical Systems . COMPOSITION OPERATOR See Also: composition operator Given a function ''g'', the composition operator is defined as that Operator which maps functions to functions as : Composition operators are studied in the field of Operator Theory . ALTERNATIVE NOTATION In the mid- 20th Century , some mathematicians decided that writing "''g'' o ''f''" to mean "first apply ''f'', then apply ''g''" was too confusing and decided to change notations. They wrote "''xf''" for "''f''(''x'')" and "''xfg''" for "''g''(''f''(''x''))". However, this movement never caught on, and nowadays this notation is found only in old books. Category Theory uses ''f;g'' interchangeably with ''g'' o ''f''. SEE ALSO
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