FUNCTION

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed set, such as the real numbers (), although different inputs may have the same output.
There are many ways to give a function: by a formula, by a plot or graph, by an algorithm that computes it, or by a description of its properties. Sometimes, a function is described through its relationship to other functions (see, for example, inverse function). In applied disciplines, functions are frequently specified by their tables of values or by a formula. Not all types of description can be given for every possible function, and one must make a firm distinction between the function itself and multiple ways of presenting or visualizing it.
One idea of enormous importance in all of mathematics is composition of functions: if z is a function of y and y is a function of x, then z is a function of x. We may describe it informally by saying that the composite function is obtained by using the output of the first function as the input of the second one. This feature of functions distinguishes them from other mathematical constructs, such as numbers or figures, and provides the theory of functions with its most powerful structure.