The concept of function with example
We
are concerned in most applications not with one variable, but with several
variables at once.
Let
us take the example of a certain quantity of air, say 1 kg; the variables
defining its state are: its pressure p(kg/m^2),
the volume v (m^3)which it occupies; its temperature t°C.
Let us assume for the moment
that the temperature of the air is maintained at 0°C; the number t
is then a constant, equal to zero. The only remaining variables
are p and
v. If the pressure p
changes, then the volume v changes;
for example, if the air is compressed, the volume decreases. We can change p
arbitrarily (at least within the limits technically attainable),
in which case we can refer to p as
an independent variable;
for every pressure p, there
is evidently a completely defined volume. There must thus be a law which enables
the corresponding volume v to
be found for every value of p. This
is, of course, Boyle's law, which says that the volume occupied by a gas at
constant temperature is
inversely proportional to the pressure. Applying this law to our kilogram of
air, the relationship between v and
p can be put in the form of an equation:
The variable v
is in this case called a function
of the independent variable p.
Turning from this particular example, we can say that, theoretically speaking, an
independent variable is characterized by a large number of possible values, its
value being any one chosen arbitrarily from all these possible values. The
independent variable x, for
example, can have
a set of values consisting of the interval (a,
b), or the interior of
this interval, i.e. the independent variable x
can take any value
satisfying the condition a <
x < b,
or a <= x <= 6. It might be
the case that x takes
any integral value, etc. In the example quoted
above, p had
the role of independent variable, and the volume v
was a
function of p. We
shall now define a function theoretically.
DEFINITION: A quantity y is called a
function of the independent variable x, if for any given value of x (from all
its possible values) there corresponds a definite value of v. Thus,
if y is
a function of x, defined
in the interval (a, b), this means that there is a corresponding definite value
of y for
any value of x from
this interval.
the source:
A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.
By: Fady tarek
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