Function composition

For function composition in computer science, see function composition (computer science).

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

In mathematics, function composition is the application of one function to the results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x.

Thus one obtains a composite function g ∘ f: X → Z defined by (g ∘ f)(x) = g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f", or "g composed with f", "g after f", "g following f", or just "g of 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 ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. 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 ∘ f = f ∘ 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 $x \ge 0$.

Considering functions as special cases of relations (namely functional relations), one can analogously define composition of relations, which gives the formula for $g \circ f \subseteq X \times Z$ in terms of $f \subseteq X \times Y$ and $g \subseteq Y \times Z$.

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.

The structures given by composition are axiomatized and generalized in category theory.

Example

The similarity that transforms triangle EFA into triangle ATB is the composition of a homothety H  and a rotation R, which share their centres (indicated by S in the diagram). For example, the image of A  under the rotation R  is U, which may be written R ( A ) = U.  And  H ( U ) = B  means that the mapping H transforms U into B. Thus H ( R ( A ) )  =  (H ○ R) ( A )  =  B.

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 ∘ h)(t) describes the oxygen concentration around the plane at time t.

Functional powers

If $Y \subseteq X$ then $f\colon X\rightarrow Y$ 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 $f\colon X\rightarrow X$ 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. For instance, a half iterate of a function f is a function g satisfying g(g(x)) = f(x). Another example would be that where f is the successor function, f ^r(x) = x + r. This idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow.

Iterated functions and flows occur naturally in the study of fractals and dynamical systems.

Composition monoids

Main article: Transformation monoid

Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain. Then one can form long, potentially complicated chains of these functions composed together, such as f ∘ f ∘ g ∘ f. Such long chains have the algebraic structure of a monoid, called transformation monoid or 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 forms a transformation 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 notations

• Many mathematicians omit the composition symbol, writing gf for g ∘ f.

• In the mid-20th century, some mathematicians decided that writing "g ∘ f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They write "xf" for "f(x)" and "(xf)g" for "g(f(x))". This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, where x is a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notation is called postfix notation. The order is important because matrix multiplication is non-commutative. Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.

• Mathematicians who use postfix notation may write "fg", meaning first do f then do g, in keeping with the order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists may write "f;g" for this, thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation a fat semicolon ⨟ (U+2A1F) is used for left relation composition. Since all functions are binary relations, it is correct to use the fat semicolon for function composition as well (see the article on Composition of relations for further details on this notation).

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.

See also

• Combinatory logic
• Composition of relations, the generalization to relations
• Function composition (computer science)
• Functional decomposition
• Flow (mathematics)
• Higher-order function
• Cobweb plot – a graphical technique for functional composition
• Lambda calculus
• Functional square root

External links

• "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.