The idea of a function. Suppose that x and y are two continuous real variables, which we may suppose to be represented geometrically by distances A0P = x, B0Q = y measured from fixed points A0, B0 along two straight lines Λ, M. And let us suppose that the positions of the points P and Q are not independent, but are connected by a relation which we can imagine expressed as a relation between x and y; so that, when P and x are known, Q and y are also known. We might, for example, suppose that y = x, or 2x, or ½x, or x2 + 1. In all of these cases the value of x determines that of y. Or again we might suppose that the relation between x and y is given, not by means of an explicit formula for y in terms of x, but by means of a geometrical construction which enables us to determine Q when P is known.

In these circumstances y is said to be a function of x. This notion of functional dependence of one variable upon another is perhaps the most important in the whole range of higher mathematics. In order to enable the reader to be certain that he understands it clearly, we shall, in this chapter, illustrate it by means of a large number of examples.