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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.[3][note 3] In other words, 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. Formally speaking, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph. Functions are also called maps or mappings, though some authors make some distinction between "maps" and "functions" (see section #Map).
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. Especially in the context of numbers, one says also that y is the value of f for the value x of its variable, or, still shorter, 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. Formally, f = g if f(x) = g(x) for all x ? X, where f:X Y and g:X Y.[4][5][note 4]
