Functions

The concept of a function is fundamental in analysis, but it is not easy to give a precise definition of it. Clearly it deals with the set of values of a variable у when another variable x takes certain values. Consider, for example, the two functions:

(i) the function whose value is 1 when x≥ 0 and 0 when x< 0;

(ii) the function whose value is 1 when x is rational and 0 when x is irrational.

The set of values of each of these functions is the finite set containing the two numbers 0 and 1; but the two functions are quite different from each other. The sets of values of the two functions x³ and x⁵ are identical (in this case the set of all real numbers), but the functions are not the same.

The essential feature of the definition of a function is the concept of a "correspondence" or "relationship" between the individual members of two sets. This correspondence is known as "many-one", that is if x denotes any member of one set and у any member of the other, then to one value of у there may correspond one, or several, or even infinitely many values of x. The student may have encountered one-one correspondence in geometry; this is a special case of many-one correspondence.

Definition: If to each member x of a certain set M there corresponds one value of a variable y, then у is said to be a function of the variable x. The variable x is called the argument of the function, and the set M the domain of the function. The set of all the values taken by the variable у is called the ordinate set. Both the domain and the ordinate set may be either finite or infinite, bounded or unbounded.

It is important to observe that it is not implicit in the definition of a function that there should exist an algebraic equation connecting x and y. If у and x are related so that у is a function of x, it does not necessarily follow that x is a function of y, although this may sometimes be true. For let X be the domain of this function and Y the ordinate set. Then if x is any member of X, we know that there is just one member у of Y which corresponds to it.

But if у is a member of Y, there may be more than one value of x in X which gives rise to a particular number, as the correspondence is many-one. If there are any values of У for which this is so, then x is not a function of у according to the definition.

Functions may be represented geometrically. For this we take a rectangular system of Cartesian coordinates in a plane and associate with each member x of the domain of the function the point P whose coordinates are (x, y). The set of points P is called the graph of the function. A function defined by means of a formula may have its domain restricted by the character of the formula itself.