Function (mathematics)

For each input value x in the domain, the corresponding unique output value y in the codomain is denoted by f(x).

A more concise expression of the above definition is the following: a function from X to Y is a subset f of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in f.

The set of all functions f : X → Y is denoted by Y^X. Note that |Y^X| = |Y|^|X| (refer to Cardinal numbers).

A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated.

Consider the following three examples:

	This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function.
	This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function.
	This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly as f = {(1, d), (2, d), (3, c)} or as :

Domains, codomains, and ranges

X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values.

Functions are named after their ranges, for example real functions and complex functions.

An endofunction is a function whose domain and range are identical.

In computer science, the datatypes of the arguments and return values specify the domain and codomain (respectively) of a subprogram. So the domain and codomain are constraints imposed initially on a function; on the other hand the range has to do with how things turn out in practice.

Injective, surjective and bijective functions

Several properties of functions that are very useful have special names:

Images and preimages

The image of an element x∈X under f is the output f(x).

The image of a subset A⊂X under f is the subset of Y formally defined by
:f[A] = {f(x) | x in A}
Usually (when subsets of the domain are not at the same time elements of the domain) one writes f(A) instead of f[A].

(An old-fashioned notation writes f'x instead of f(x), and f"A instead of f[A].)
Notice that the range of f is the image f(X) of its domain. In our function above, the image of {2,3} under f is f({2, 3}) = {c, d} and the range of f is {a, c, d}.

The preimage (or inverse image) of a set B ⊂ Y under f is the subset of X defined by
:f ⁻¹(B) = {x in X | f(x) ∈B}
For our function above, the preimage of {a, b} is f ⁻¹({a, b}) = {1}.

Graph of a function

The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem. If X and Y are real lines, then this definition coincides with the familiar sense of graph.

, This function is surjective but not injective.

Note that a binary relation on the two sets X and Y could be identified with an orderred triple (X,Y,G) where G is the graph of the relation or
identified with the its graph. In the latter case a function f is the same as its graph.

Examples of functions

(More can be found at list of functions.)

• The relation wght between persons in the United States and their weights at a particular time.
• The relation between nations and their capitals, if we exclude those nations that maintain multiple capitals [1].
• The relation sqr between natural numbers n and their squares n².
• The relation ln between positive real numbers x and their natural logarithms ln(x). Note that the relation between real numbers and their natural logarithms is not a function because not every real number has a natural logarithm; that is, this relation is not total.
• The relation dist between points in the plane R² and their distances from the origin (0,0).
• The relation grav between a point in the punctured plane R² \\ {(0,0)} and the vector describing the gravitational force that a certain mass at that point would experience from a certain other mass at the origin (0,0).

Properties of functions

Ambiguous functions

Strictly speaking, an ambiguous function is not truly a function because a mathematical function is defined as having "a unique output to each given input". In fact, such "functions" are more properly termed relations.

n-ary function: function of several variables

Functions in applications are often functions of several variables, or multivariate functions: the values they take depend on a number of different factors. From a mathematical point of view all the variables must be made explicit in order to have a functional relationship - no 'hidden' factors are allowed. Then again, from the mathematical point of view, there is no qualitative difference between functions of one and of several variables. A function of three real variables is just a function that applies to triples of real numbers. The following paragraph says this in more formal language.

If the domain of a function is a subset of the Cartesian product of n sets then the function is called an n-ary function.
For example, the relation dist has the domain R × R and is therefore a binary function.
In that case dist((x,y)) is simply written as dist(x,y).

Another name applied to some types of functions of several variables is operation. In abstract algebra, operators such as "*" are defined as binary functions; when we write a formula such as x*y in this context, we are implicitly invoking the function *(x,y), but writing it in a convenient infix notation.

An important theoretical paradigm, functional programming, takes the function concept as central. In that setting, the handling of functions of several variables becomes an operational matter, for which the lambda calculus provides the basic syntax. The composition of functions (see under composing functions immediately below) becomes a question of explicit forms of substitution, as used in the substitution rule of calculus. In particular, a formalism called currying can be used to reduce n-ary functions to functions of a single variable.

Composing functions

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 composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X.
As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x).
Then (c o h)(t) describes the oxygen concentration around the plane at time t.

Inverse function

An example of an inverse function, for f(x) = 2x, is f(x)⁻¹ = x/2. The inverse function is the function that "undoes" its original. See also inverse image.

Inverses are sometimes difficult or impossible to find. Consider f(x) = x². The function f(x) = √x is not an inverse when the domain of f is R. (As -2² is 4, but √4 is either 2 or -2).

Restrictions and extensions

The restriction of to is then the function . Intuitively, this is the same function as except that we restrict the domain of to .

An extension of a function is a function defined on a superset of such that . Provided the domain of is not the universal set, always has lots of extensions.

Pointwise operations

If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows:
:(f + g)(x) = f(x) + g(x)
:(f × g)(x) = f(x) × g(x)
for all x in X.

This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.

By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way.

Computable and non-computable functions

The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. This argument shows that there are functions from integers to integers that are not computable. For examples of noncomputable functions, see the articles on the halting problem and Rice's theorem.

Functions from the categorical viewpoint

In the formal definition, a function represents a relationship between its domain and its codomainn, rather than just a rule for taking an input to an output. A generalisation of the notion of funtion is morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection.

Ordinary functions are sometimes referred to as morphisms in a concrete category.

References

See also

External links