Function composition
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In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. 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 bijective functions f: X → X form a group 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 x, the first expression is undefined. (But inverse functions always commute to produce the identity mapping.)
Derivatives 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.
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[edit] 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.
[edit] 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 n o f = f n+1 for natural n follow immediately.
- By convention, f 0 = idD(f) (the identity map on the domain of f).
- If f:X→X admits an inverse function, negative functional powers f -k (k > 0) are defined as the opposite power of the inverse function, (f −1)k.
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.
[edit] Composition operator
Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as
Composition operators are studied in the field of operator theory.
[edit] 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 usually only found in old books.
Category Theory uses f;g interchangeably with g o f.
