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Example As an example, suppose that an airplane's elevation at time t is given by the function h(t) and that the oxygen concentration at elevation 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: X → Y 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 called function iteration. The functional powers f o f n f 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 functions: 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. A simple example would be that where f is the successor function, f r(x) = x + r. Iterated functions occur naturally in the study of fractals and dynamical systems. 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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