, a function is originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of the time. Historically, the concept was elaborated with the infinitesimal calculus at the end of 17th century, and, until the 19th century, the functions that were considered were differentiable (that is they have a high degree of regularity). The concept of function has been formalized at the end of the 19th century in terms of set theory, and this has much 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 says often that f is a function of the variable x).
A function is uniquely represented by its graph which is the set of all pairs (x, f (x)). 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 in newspapers.
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. (Wikipedia)
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