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(More can be found at List of functions.)

The relation wght between persons in the United States and their weights at a particular time.

The relation between nations and their capitals, if we exclude those nations that maintain multiple capitals .

The relation sqr between natural numbers n and their squares n2.

The relation ln between positive real numbers x and their natural logarithms ln(x). Note that the relation between réal numbers and their natural logarithms is not a function because not every réal number has a natural logarithm; that is, this relation is not total.

The relation dist between points in the plane R2 and their distances from the origin (0,0).

The relation grav between a point in the punctured plane R2 \ {(0,0)} and the vector describing the gravitational force that a certain mass at that point would experience from a certain other mass at the origin (0,0).

Most commonly used types of mathematical functions involving addition, division, exponents, logarithms, multiplication, polynomials, radicals, rationals, subtraction, and trigonometric expressions. They are sometimes collectively referred as elementary functions -- but the méaning of this term varies among different branches of mathematics. Example of non-elementary functions (or special functions) are Bessel functions sarta fungsi gamma.

n-ary function: function of several variables

Functions in applications are often functions of several variables: the values they take depend on a number of different factors. From a mathematical point of view all the variables must be made explicit in order to have a functional relationship - no 'hidden' factors are allowed. Then again, from the mathematical point of view, there is no qualitative difference between functions of one and of several variables. A function of three réal variables is just a function that applies to triples of réal numbers. The following paragraph says this in more formal language.

If the domain of a function is a subset of the Cartesian product of n sets then the function is called an n-ary function.

For example, the relation dist has the domain R × R and is therefore a binary function.

In that case dist((x,y)) is simply written as dist(x,y).

Another name applied to some types of functions of several variables is operation. In abstract algebra, operators such as "*" are defined as binary functions; when we write a formula such as x*y in this context, we are implicitly invoking the function *(x,y), but writing it in a convenient infix notation.

An important théoretical paradigm, functional programming, takes the function concept as central. In that setting, the handling of functions of several variables becomes an operational matter, for which the lambda calculus provides the basic syntax. The composition of functions (see under composing functions immediately below) becomes a question of explicit forms of substitution, as used in the substitution rule of calculus. In particular, a formalism called currying can be used to reduce n-ary functions to functions of a single variable.