Function composition

Related subjects: Mathematics

Background Information

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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", "g following f", or just "g of f".

g o f, the composition of f and g

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.

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.

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.

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.

The functional powers $f\circ f^n=f^n\circ f=f^{n+1}$ for natural $n\,$ follow immediately.

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, $(f^{-1})^k\,$ .

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 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 general, composition monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup on X.

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.

The set of all bijective functions f: X → X form a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group.

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.

Category Theory uses f;g interchangeably with g o f.

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.

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