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". Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

g o f, the composition of f and g

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. Image File history File links The figure of a composite function -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links The figure of a composite function -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the...

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. In mathematics, associativity is a property that a binary operation can have. ...

As a result the set of bijective functions f: X → X form a group with respect to the composition operator. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

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. Only the simplest functions will be commutative under composition. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

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. In mathematics, the derivative is one of the two central concepts of calculus. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... // The formula Faà di Brunos formula is an identity in mathematics generalizing the chain rule to higher derivatives, named in honor of Francesco Faà di Bruno (1825–1888), who was (in chronological order) a military officer, a mathematician, and a priest, and was beatified by the Pope a century...

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)

The functional powers f o f ⁿ = f ⁿ o f = f ⁿ⁺¹ for natural n follow immediately. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

• By convention, f ⁰ = id_D(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). In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the word opposite in some contexts means additive inverse; it can also mean an opposite category. ... A ring is usually anything resembling a circle, or a noise that cycles rapidly. ...

(For usual numerical 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 mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... It has been suggested that this article or section be merged with exponential function. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

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 functions occur naturally in the study of fractals and dynamical systems. In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...

See also

• combinator
• lambda calculus
• higher-order function
• Function composition (computer science)