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Encyclopedia > Function composition

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 fX → Y and gY → 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". Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

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 go h. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. 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 mathematics, associativity is a property that a binary operation can have. ...

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, $left | x right | + 3 = left | x + 3 right |,$ only when $x ge 0$. But inverse functions always commute to produce the identity mapping. 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. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

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. For a non-technical overview of the subject, see Calculus. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... // The formula FaÃ  di Brunos formula is an identity in mathematics generalizing the chain rule to higher derivatives, named in honor of Francesco FaÃ  di Bruno (1825â€“1888), who was (in chronological order) a military officer, a mathematician, and a priest, and was beatified by the Pope a century...

## Contents

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 $Y subseteq X$ then $f: Xrightarrow Y$ may compose with itself; this is sometimes denoted $f^2,$. Thus: A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... $(fcirc f)(x) = f(f(x)) = f^2(x)$ $(fcirc fcirc f)(x) = f(f(f(x))) = f^3(x)$

Repeated composition of a function with itself is called function iteration. In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ...

The functional powers $fcirc f^n=f^ncirc f=f^{n+1}$ for natural $n,$ follow immediately. Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

• By convention, $f^0= id_{D(f)},$ $big($the identity map on the domain of $fbig)$.
• If $f: Xrightarrow X$ 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 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. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

(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 mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

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. A successor function is the label in the literature for what is actually an operation. ...

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. ...

## Composition monoids

Suppose one has two (or more) functions f: XX, g: XX having the same domain and range. Then one can form long, potentialy complicated chains of these functions composed together, such as f o f o g o f. Such long chains have the algebraic structure of a monoid, sometimes called the composition monoid. In general, composition monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: XX is called the full transformation semigroup on X. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a de Rham curve is a certain type of fractal curve. ... It has been suggested that state transition system be merged into this article or section. ...

If the functions are bijective, then the set of all possible combinations of these functions form a group; and one says that the group is generated by these functions. 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. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

The set of all bijective functions fX → X form a group with respect to the composition operator; this is sometimes called the composition group. 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. ...

## 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))". This can be more natural and seem simpler than writing functions on the left in some areas. (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&#8211;2000 in the sense of the Gregorian calendar (1900&#8211;1999...

Category Theory uses f;g interchangeably with g o f. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

## Composition operator

Main article: composition operator

Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as In mathematics, the composition operator with symbol is defined by the rule where denotes function composition. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... $C_g f = f circ g$

Composition operators are studied in the field of operator theory. In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. ...

Combinatory logic is a notation introduced by Moses SchÃ¶nfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. ... In computer science, function composition is an act or mechanism to combine simple functions to build more complicated ones. ... Functional decomposition of engineering is a method for analyzing engineered systems. ... In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... In logic and mathematics, the composition of relations is the generalization of the composition of functions. ... Results from FactBites:

 Composition of Functions in Math. How to evaulate and express a composition of two functions (609 words) The composition of function notation on the left is the composition. Pictured on the right is a diagram of the role of the domain and range of each function in the composition of f(g(x)). Compositions of Functions are a well known example of a process that is not commutative.
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