 # function

A function, in mathematics, associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can also be elements from any given set. An example of a function is f(x) = 2x, a function which associates with every number the number twice as large. Thus 5 is associated with 10, and this is written f(5) = 10.

The input to a function need not be a number, it can be any well defined object. For example, a function might associate the letter A with the number 1, the letter B with the number 2, and so on. There are many ways to describe or represent a function, such as a formula or algorithm that computes the output for a given input, a graph that gives a picture of the function, or a table of values that gives the output for certain specified inputs. Tables of values are especially common in statistics, science, and engineering.

The set of all inputs to a particular function is called the domain?. In modern mathematics functions are normally defined to have a codomain? associated with them which is some fixed set which includes all possible outputs, for instance real valued functions have a codomain? which includes all the real numbers even though each particular real valued function may not include every real number as an output. The set of all the ordered pairs or inputs and outputs (x, f(x)) of a function is called its graph. A common way to define a function is as the triple (domain?, codomain?, graph), that is as the input set, the possible outputs and the mapping for each input to its output.

The set of all outputs of a particular function is called its image. The word range is used in some texts to refer to the image and in others to the codomain?, in particular in computing it often refers to the codomain?. The domain? and codomain? are often "understood". Thus for the example given above, f(x) = 2x, the domain? and codomain? were not stated explicitly. They might both be the set of all real numbers, but they might also be the set of integers?. If the domain? is the set of integers?, then image consists of just the even integers?.

There are many ways to describe or represent functions: for instance by a formula, by an algorithm? that computes it, or simply by enumerating its values for every possible argument. A function may also be described through its relationship to other functions, for example, as the inverse function of a given function, or as a solution of a differential equation. Functions can be added, multiplied, or combined in other ways to produce new functions. An important operation on functions, which distinguishes them from numbers, is the composition of functions. There are uncountably many different functions, most of which cannot be expressed with a formula or an algorithm?.

Collections of functions with certain properties, such as continuous functions and differentiable functions, are called function spaces and are studied as objects in their own right, in such mathematical disciplines as real analysis and complex analysis.

Calculus
Continuity
fundamental theorem of calculus
Laplacian
Limit of a Function
Stokes Theorem