Variables and functions

Variables and constants
A variable is a quantity to which an unlimited number of values can be assigned.
Variables are denoted by the later letters of the alphabet. Thus, in the equation
of a straight line,
xa+yb= 1
x and y may be considered as the variable coordinates of a point moving along
the line. A quantity whose value remains unchanged is called a constant.
Numerical or absolute constants retain the same values in all problems, as 2,
5, √7, π, etc.
Arbitrary constants, or parameters, are constants to which any one of an
unlimited set of numerical values may be assigned, and they are supposed to have
these assigned values throughout the investigation. They are usually denoted
by the earlier letters of the alphabet. Thus, for every pair of values arbitrarily
assigned to a and b, the equation
xa+yb= 1
represents some particular straight line.

Interval of a variable.
Very often we confine ourselves to a portion only of the number system. For
example, we may restrict our variable so that it shall take on only such values as
lie between a and b, where a and b may be included, or either or both excluded.
We shall employ the symbol [a, b], a being less than b, to represent the numbers
a, b, and all the numbers between them, unless otherwise stated. This symbol
[a, b] is read the interval from a to b.

CONTINUOUS VARIATION.
Continuous variation.
A variable x is said to vary continuously through an interval [a, b], when x
starts with the value a and increases until it takes on the value b in such a
manner as to assume the value of every number between a and b in the order of
their magnitudes. This may be illustrated geometrically as follows:
Figure 2.1: Interval from A to B.
The origin being at O, layoff on the straight line the points A and B correspond-
ing to the numbers a and b. Also let the point P correspond to a particular
value of the variable x. Evidently the interval [a, b] is represented by the seg-
ment AB. Now as x varies continuously from a to b inclusive, i.e. through the
interval [a, b], the point P generates the segment AB.

Functions.
When two variables are so related that the value of the first variable depends on
the value of the second variable, then the first variable is said to be a function
of the second variable.
Nearly all scientific problems deal with quantities and relations of this sort,
and in the experiences of everyday life we are continually meeting conditions
illustrating the dependence of one quantity on another. For instance, the weight
a man is able to lift depends on his strength, other things being equal. Similarly,
the distance a boy can run may be considered as depending on the time. Or,
we may say that the area of a square is a function of the length of a side, and
the volume of a sphere is a function of its diameter.