An introduction to mathematics, by A. N. Whitehead.

FUNCTIONS 147
F(x), f(x), g(x), (x), etc., for any function of
x, where the argument x is placed in a bracket
and some letter like F, f, g, q, 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, b, 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. Q 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.
With these explanations and cautions, we
write y =f(x), to denote that y is the value of
some undetermined function of the argument
x; where f(x) may stand for anything such
as x+1, x2 -x +1, sin x, log x, or merely for
x itself. The essential point is that when z
is given, then y is thereby definitely determined. It is important to be quite clear as
to the generality of this idea. Thus in y=
f(x), we may determine, if we choose, f(x) to
mean that when x is an integer, f(x) is zero,
and when x has any other value, f(x) is 1.
Accordingly, putting y -f(x), with this choice