Function composition

For function composition in computer science, see function composition (computer science). In Computer science, function composition (not to be confused with Object composition) is an act or mechanism to combine simple functions to build more

g o f, the composition of f and g. For example, (g o f)(c) = #.

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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", "g following f", or just "g of f".

The composition of functions is always associative. In Mathematics, associativity is a property that a Binary operation can have 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.

The functions g and f are said to commute with each other if g o f = f o g. In Mathematics, commutativity is the ability to change the order of something without changing the end result 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, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that

Derivatives of compositions involving differentiable functions can be found using the chain rule. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. "Higher" derivatives of such functions are given by Faà di Bruno's formula. Faà di Bruno's formula is an identity in Mathematics generalizing the Chain rule to higher derivatives named in honor of Francesco Faà di Bruno (1825&ndash1888

1 Example
2 Functional powers
3 Composition monoids
4 Alternative notation
5 Composition operator
6 See also
7 External links

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 $Y \subseteq X$ then $f: X\rightarrow Y$ may compose with itself; this is sometimes denoted $f^2\,$ . Thus:

$(f\circ f)(x) = f(f(x)) = f^2(x)$

$(f\circ f\circ f)(x) = f(f(f(x))) = f^3(x)$

Repeated composition of a function with itself is called function iteration. In Mathematics, iterated functions are the objects of deep study in Computer science, Fractals and Dynamical systems An iterated function is

The functional powers $f\circ f^n=f^n\circ f=f^{n+1}$ for natural $n\,$ follow immediately. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

By convention, $f^0= id_{D(f)}\,$ $\big($ the identity map on the domain of $f\big)$ .
If $f: X\rightarrow X$ admits an inverse function, negative functional powers $f^{-k}\,$ $(k>0\,)$ are defined as the opposite power of the inverse function, $(f^{-1})^k\,$ . In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero.

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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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). In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables 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. The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict Subset of the recursive

Iterated functions occur naturally in the study of fractals and dynamical systems. In Mathematics, iterated functions are the objects of deep study in Computer science, Fractals and Dynamical systems An iterated function is A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential

Composition monoids

Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and range. Then one can form long, potentially 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 Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In general, composition monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. In Mathematics, a de Rham curve is a certain type of Fractal Curve. The set of all functions f: X → X is called the full transformation semigroup on X. In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output

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, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the

The set of all bijective functions f: X → X form a group with respect to the composition operator. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property This is the symmetric group, also sometimes called the composition group. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

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. The twentieth century of the Common Era began on 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.

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 it abstracts from sets

Composition operator

Main article: composition operator

Given a function g, the composition operator $C g$ is defined as that operator which maps functions to functions as

$C_g f = f \circ g.$

Composition operators are studied in the field of operator theory. In Mathematics, the composition operator C_\phi with symbol \phi is defined by the rule C_\phi (f = f \circ\phi In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics, operator theory is the branch of Functional analysis which deals with Bounded linear operators and their properties

External links

"Composition of Functions" by Bruce Atwood, The Wolfram Demonstrations Project, 2007. Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for Variables in Mathematical logic In Computer science, function composition (not to be confused with Object composition) is an act or mechanism to combine simple functions to build more Functional decomposition refers broadly to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed In Mathematics and Computer science, higher-order functions or '''functionals''' are functions which do at least one of the following In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function In Mathematics, the composition of Binary relations is a concept of forming a new relation S o R from two given relations R and S, having as

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents