Everything about Function Composition totally explained
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".
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's 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 won't be commutative. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example,
only when
. 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
then
may compose with itself; this is sometimes denoted
. Thus:
»
»
Repeated composition of a function with itself is called
function iteration.
The
functional powers
.
Note: If
f takes its values in a
ring (in particular for real or complex-valued
f ), there's a risk of confusion, as
f n could also stand for the
n-fold product of
f, for example
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:
sin
2(
x) = sin(
x) · sin(
x).
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, for example, 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 is defined as that
operator which maps functions to functions as
»
Composition operators are studied in the field of
operator theory.
Further Information
Get more info on 'Function Composition'.
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