Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/32

8 The symbol $$\scriptstyle{f(x)}$$ is used to denote a function of $$\scriptstyle{x}$$, and is read $$\scriptstyle{f}$$ of $$\scriptstyle{x}$$. In order to distinguish between different functions, the prefixed letter is changed, as $$\scriptstyle{F(x),~\phi(x),~f'(x)}$$, etc.

During any investigation the same functional symbol always indicates the same law of dependence of the function upon the variable. In the simpler cases this law takes the form of a series of analytical operations upon that variable. Hence, in such a case, the same functional symbol will indicate the same operations or series of operations, even though applied to different quantities. Thus, if

Similarly, $$\scriptstyle{\phi(x,y)}$$ denotes a function of $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ and is read $$\scriptstyle{\phi}$$ of $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$.

Evidently this system of notation may be extended indefinitely.

Consider the functionsof the independent variable $$\scriptstyle{x}$$. Denoting the dependent variable in each case by $$\scriptstyle{y}$$, we may write