In mathematics, function is a map from a input set to a output set.

The input set is called **domain**, the output set is called **codomain.**

A map is a function if every element in the domain can be mapped to one and only one element in the codomain.

Set of all actual output elements is called range, range could be a subset of codomain, i.e. a real function f: x->x*x, x is an element in the set of real number, the codomain is also the set of real number because f is a real function, but the range is the set of all non-negative real numbers, because x*x cannot be negative.

Some functions can be stateful, i.e. a counter function generates incremental number sequence, every time you call it, it returns a different number, because it saves a state of current number inside. In this case, you can either consider the concept is different mathematics, or you can treat “states” as a part of function input.

In mathematics, a function is a relation between sets, that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).

A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are “the central objects of investigation” in most fields of mathematics.

**Types of Functions**

Based on the kind of element that the sets involved consists of functions can be:

- Identity function: y=f(x)=x; both range and domain of the function is the same.
- Constant function: y=f(x)=Constant; the range of the function is constant
- Polynomial function: y=f(x)=polynomial for every value of x.
- Rational function: these are y=f(x)=g(x)/h(x) type of function where both g(x) and h(x) are polynomials and h(x)≠0.
- Modulus function: The range of the function is positive plus the set of 0
- Signum function: Range of f is {-1, 0, 1}
- Greatest Integer function: The function assumes the value of the greatest integer. The range is a pure integer value.

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