Function (mathematics) - Notation

Notation

A function f with domain X and codomain Y is commonly denoted by

or

In this context, the elements of X are called arguments of f. For each argument x, the corresponding unique y in the codomain is called the function value at x or the image of x under f. It is written as f(x). One says that f associates y to x or maps x to y. This is abbreviated by

A general function is often denoted by f. If a function is often used, it may be given a special name as, for example, the signum function of a real number x is denoted by sgn(x). The argument is often denoted by the symbol x, but in other contexts may be denoted differently, as well. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). It is common to omit the parentheses around the argument when there is little chance of confusion, thus: sin x; this is known as prefix notation.

In order to specify a concrete function, the notation (an arrow with a bar at its tail) is used. For example, the above function reads

The first part is read:

• "f is a function from (the set of natural numbers) to (the set of integers)" or
• "f is an -valued function of an -valued variable".

The second part is read "n maps to 4−n." In other words, this function has the natural numbers as domain, the integers as codomain. A function is properly defined only when the domain and codomain are specified. For example, the formula f(x) = 4 − x alone (without specifying the codomain and domain) is not a properly defined function. Moreover, the function

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x)=4-x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. For instance, stands for the function, stands for the integral function, and so on.