An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

FUNCTIONS.

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tative of the particular operation or set of operations performed
upon x. Thus if f(x) = xn, f represents the operation of
taking a quantity to the nth power; if f(x) = log sin, f
represents the operations of taking the logarithm of the sine
of a quantity; if BPQ (fig. 1) be a given curve, AM( x)
MP (= y) the co-ordinates of any point P, and if we take
f(x) to represent y, then f represents the operations of measuring AM equal to x along the line AX, erecting MP perpendicular to AX, producing it to meet the curve at P, and so
finding the length MP or y.
5. From the definition here given of a function we may A function
see that f (x) is not necessarily a quantity which changes when iSe Ots
x changes; for a set of operations performed upon x may a quantity
sometimes lead to a result which is the same for all values changes
when the
of iZ. This will appear from the following example. variable
changes.
Let APB (fig. 2) be a semicircle whose radius is a, C its
center; take any arc AP = I, with P as center, and some
given line c as radius, describe a circle cutting AB at the
point Q: then we may say that AQ is a function of x and
denote it by f(x); f will therefore represent the operations
of measuring x along AP, describing a circle with radius c
and center P, so finding Q, and therefore finally AQ.
Now in general AQ or f(x) will be different for different
values of x; but if we take c = a, then the circle described with
P as center and c (= a) as radius will always cut AB at the
center C, and therefore we shall have AQ = AC or f(x) = a,
whatever be the value of x. Hence f (x) in this case does not
vary when x varies.
Thus it appears, according to our definition of a function,
that f (x) is not necessarily a quantity which changes when x
changes; and this remark is important, as will appear hereafter.
6. When we have occasion to consider several different
functions at the same time, we employ different functional
letters, in order to distinguish between them. The letters
commonly used, in addition to f, are F, c, Al, X, and sometimes these letters with dashes, thus f', F, p(', 'j, X' or f"
F", &c...&c. Thus we might put f (I) to represent x", q (~)
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