XI

Functions

The mathematical use of the term function

has been adopted also in common life. For example, "His temper is a function of his digestion," uses the term exactly in this mathematical sense. It means that a rule can be assigned which will tell you what his temper will be when you know how his digestion is working. Thus the idea of a "function" is simple enough, we only have to see how it is applied in mathematics to variable numbers. Let us think first of some concrete examples: If a train has been travelling at the rate of twenty miles per hour, the distance ($s$ miles) gone after any number of hours, say $t$, is given by $s=20\times t$; and $s$ is called a function of $t$. Also $20\times t$ is the function of $t$ with which $s$ is identical. If John is one year older than Thomas, then, when Thomas is at any age of $x$ years, John's age ($y$ years) is given by $y=x+1$; and $y$ is a function of $x$, namely, is the function $x+1$.

In these examples $t$ and $x$ are called the

"arguments" of the functions in which they appear. Thus $t$ is the argument of the function $20\times t$, and $x$ is the argument of the function $x+1$. If $s=20\times t$, and $y=x+1$, then $s$ and $y$ are called the "values" of the functions $20\times t$ and $x+1$ respectively.

Coming now to the general case, we can define a function in mathematics as a correlation between two variable numbers, called respectively the argument and the value of the function, such that whatever value be assigned to the "argument of the function" the "value of the function" is definitely (i.e. uniquely) determined. The converse is not necessarily true, namely, that when the value of the function is determined the argument is also uniquely determined. Other functions of the argument $x$ are $y=x^2$,

$y=2x^2+3x+1$, $y=x$, $y=\log x$, $y=\sin x$. The last two functions of this group will be readily recognizable by those who understand a little algebra and trigonometry. It is not worth while to delay now for their explanation, as they are merely quoted for the sake of example.

Up to this point, though we have defined what we mean by a function in general, we have only mentioned a series of special functions. But mathematics, true to its general methods of procedure, symbolizes the general idea of any function. It does this by writing

$F(x)$, $f(x)$, $g(x)$, $\varphi (x)$, etc., for any function of $x$, where the argument $x$ is placed in a bracket, and some letter like $F$, $f$, $g$, $\varphi$, etc., is prefixed to the bracket to stand for the function. This notation has its defects. Thus it obviously clashes with the convention that the single letters are to represent variable numbers; since here $F$, $f$, $g$, $\varphi$, etc., prefixed to a bracket stand for variable functions. It would be easy to give examples in which we can only trust to common sense and the context to see what is meant. One way of evading the confusion is by using Greek letters (e.g. $\varphi$ as above) for functions; another way is to keep to $f$ and $F$ (the initial letter of function) for the functional letter, and, if other variable functions have to be symbolized, to take an adjacent letter like $g$.