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A function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set

Functions were originally the idealization of how a varying quantity depends on another quantity

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, the codomain of the function

A function is usually denoted by letters such as f, g and h

If the function is called f, this relation is denoted by y = f(x), where 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 (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 real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane; the set of these points is called the graph of the function

A function is a process that associates each element of a set X, to a single element of a set Y

Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x,y) such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G

For every x in X, there is exactly one element y such that the ordered pair (x,y) belongs to the set of pairs defining the function f; the set G is called the graph of the function

In the definition of function, X and Y are respectively called the domain and the codomain of the function f

If (x,y) belongs to the set defining f, then y is the image of x under f, or the value of f applied to the argument x

In the context of numbers in particular, one also says that y is the value of f for the value x of its variable, or, more concisely, that y is the value of f of x, denoted as y = f(x)

Two functions f and g are equal, if their domain and codomain sets are the same and their output values agree on the whole domain; more formally, f = g if f(x) = g(x) ∀ x ∈ X, where f: X → Y and g: X → Y

A = domain of the function, B = codomain of the function; f: A → B, a ∈ A → b = f(a) ∈ B

Image of the function: Im(f) = {b ∈ B | ∃ a ∈ A | b = f(a)} = f(A)

Graph of the function: G(f) = {(a,b) ∈ AxB | b = f(a)}