Suppose that \(x\) and \(y\) are two continuous real variables, which we may suppose to be represented geometrically by distances \(A_{0}P = x\), \(B_{0}Q = y\) measured from fixed points \(A_{0}\)\(B_{0}\) along two straight lines \(\Lambda\), M. And let us suppose that the positions of the points \(P\) and \(Q\) are not independent, but connected by a relation which we can imagine to be 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 \(y = 2x\), or \(\frac{1}{2}x\), or \(x^{2} + 1\). 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.

But before we proceed to do this, we must point out that the simple examples of functions mentioned above possess three characteristics which are by no means involved in the general idea of a function, viz.:

(1) \(y\) is determined for every value of \(x\);

(2) to each value of \(x\) for which \(y\) is given corresponds one and only one value of \(y\);

(3) the relation between \(x\) and \(y\) is expressed by means of an analytical formula, from which the value of \(y\) corresponding to a given value of \(x\) can be calculated by direct substitution of the latter.

It is indeed the case that these particular characteristics are possessed by many of the most important functions. But the consideration of the following examples will make it clear that they are by no means essential to a function. All that is essential is that there should be some relation between \(x\) and \(y\) such that to some values of \(x\) at any rate correspond values of \(y\).

Example X

1. Let \(y = x\) or \(2x\) or \(\frac{1}{2}x\) or \(x^{2} +1\). Nothing further need be said at present about cases such as these.

2. Let \(y = 0\) whatever be the value of \(x\). Then \(y\) is a function of \(x\), for we can give \(x\) any value, and the corresponding value of \(y\) (viz. \(0\)) is known. In this case the functional relation makes the same value of \(y\) correspond to all values of \(x\). The same would be true were \(y\) equal to \(1\) or \(-\frac{1}{2}\) or \(\sqrt{2}\) instead of \(0\). Such a function of \(x\) is called a constant.

3. Let \(y^{2} = x\). Then if \(x\) is positive this equation defines two values of \(y\) corresponding to each value of \(x\), viz. \(\pm\sqrt{x}\). If \(x = 0\), \(y = 0\). Hence to the particular value \(0\) of \(x\) corresponds one and only one value of \(y\). But if \(x\) is negative there is no value of \(y\) which satisfies the equation. That is to say, the function \(y\) is not defined for negative values of \(x\). This function therefore possesses the characteristic (3), but neither (1) nor (2).

4. Consider a volume of gas maintained at a constant temperature and contained in a cylinder closed by a sliding piston.1

Let \(A\) be the area of the cross section of the piston and \(W\) its weight. The gas, held in a state of compression by the piston, exerts a certain pressure \(p_{0}\) per unit of area on the piston, which balances the weight \(W\), so that \[W = Ap_{0}.\]

Let \(v_{0}\) be the volume of the gas when the system is thus in equilibrium. If additional weight is placed upon the piston the latter is forced downwards. The volume (\(v\)) of the gas diminishes; the pressure (\(p\)) which it exerts upon unit area of the piston increases. Boyle’s experimental law asserts that the product of \(p\) and \(v\) is very nearly constant, a correspondence which, if exact, would be represented by an equation of the type \[\begin{equation*} pv = a, \tag{i}\end{equation*}\] where \(a\) is a number which can be determined approximately by experiment.

Boyle’s law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When \(v\) is decreased and \(p\) increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i). It is known that a much better approximation to the true relation can then be found by means of what is known as ‘van der Waals’ law’, expressed by the equation \[\begin{equation*} \left(p + \frac{\alpha}{v^{2}}\right)(v – \beta) = \gamma, \tag{ii}\end{equation*}\] where \(\alpha\), \(\beta\), \(\gamma\) are numbers which can also be determined approximately by experiment.

Of course the two equations, even taken together, do not give anything like a complete account of the relation between \(p\) and \(v\). This relation is no doubt in reality much more complicated, and its form changes, as \(v\) varies, from a form nearly equivalent to (i) to a form nearly equivalent to (ii). But, from a mathematical point of view, there is nothing to prevent us from contemplating an ideal state of things in which, for all values of \(v\) not less than a certain value \(V\), (i) would be exactly true, and (ii) exactly true for all values of \(v\) less than \(V\). And then we might regard the two equations as together defining \(p\) as a function of \(v\). It is an example of a function which for some values of \(v\) is defined by one formula and for other values of \(v\) is defined by another.

This function possesses the characteristic (2); to any value of \(v\) only one value of \(p\) corresponds: but it does not possess (1). For \(p\) is not defined as a function of \(v\) for negative values of \(v\); a ‘negative volume’ means nothing, and so negative values of \(v\) do not present themselves for consideration at all.

5. Suppose that a perfectly elastic ball is dropped (without rotation) from a height \(\frac{1}{2}g\tau^{2}\) on to a fixed horizontal plane, and rebounds continually.

The ordinary formulae of elementary dynamics, with which the reader is probably familiar, show that \(h = \frac{1}{2}gt^{2}\) if \(0 \leq t \leq \tau\), \(h = \frac{1}{2}g(2\tau – t)^{2}\) if \(\tau \leq t \leq 3\tau\), and generally \[h = \tfrac{1}{2}g(2n\tau – t)^{2}\] if \((2n – 1)\tau \leq t \leq (2n + 1)\tau\), \(h\) being the depth of the ball, at time \(t\), below its original position. Obviously \(h\) is a function of \(t\) which is only defined for positive values of \(t\).

6. Suppose that \(y\) is defined as being the largest prime factor of \(x\). This is an instance of a definition which only applies to a particular class of values of \(x\), viz. integral values. ‘The largest prime factor of \(\frac{11}{3}\) or of \(\sqrt{2}\) or of \(\pi\)’ means nothing, and so our defining relation fails to define for such values of \(x\) as these. Thus this function does not possess the characteristic (1). It does possess (2), but not (3), as there is no simple formula which expresses \(y\) in terms of \(x\).

7. Let \(y\) be defined as the denominator of \(x\) when \(x\) is expressed in its lowest terms. This is an example of a function which is defined if and only if \(x\) is rational. Thus \(y = 7\) if \(x = -11/7\): but \(y\) is not defined for \(x = \sqrt{2}\), ‘the denominator of \(\sqrt{2}\)’ being a meaningless form of words.

8. Let \(y\) be defined as the height in inches of policeman \(Cx\), in the Metropolitan Police, at 5.30 P.M.  on 8  Aug. 1907. Then \(y\) is defined for a certain number of integral values of \(x\), viz. \(1\), \(2\), …, \(N\), where \(N\) is the total number of policemen in division \(C\) at that particular moment of time.

  1. I borrow this instructive example from Prof. H. S. Carslaw’s Introduction to the Calculus.↩︎

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