20. The idea of a function

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 $$W=A{p}_0.$$

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 $$\begin{array}{}\text{(i)}& pv=a,\end{array}$$ 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 $$\begin{array}{}\text{(ii)}& (p+\frac{\alpha }{v^2})(v–\beta )=\gamma ,\end{array}$$ 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 $h=\frac{1}{2}g{t}^2$ if $0\le t\le \tau$, $h=\frac{1}{2}g(2\tau –t{)}^2$ if $\tau \le t\le 3\tau$, and generally $$h=\frac{1}{2}g(2n\tau –t{)}^2$$ if $(2n–1)\tau \le t\le (2n+1)\tau$, $h$ 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.