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A function relates one
variable to another. Usually this is
done in terms of an equation involving one
variable on the left hand side of the equals
sign and an expression involving another
variable and other constants on the right hand
side of the equation.
Example: Y =
130 * X + 20
here we have 2 constants
130 and 20, and two variables Y and X.
Since Y is by itself on
the left hand side of the equation, we say
that Y "depends" on the value of X.
That is, X can take on new values and Y will
change accordingly (as defined by the equation).
So, Y is the "dependent" variable and X is the
"independent variable."
Generally speaking a function associates
values from one set to values of another set.
These sets can be anything whatsoever, but in
our discussions these sets will contain
numbers. We say that the function "maps"
values from one domain into another.
Using the nomenclature above,
a function maps the independent domain into
the dependent domain.
To distinguish these two ranges we use the
word "domain" for the independent
domain and "range" for the dependent
domain. So, X values come from the
domain and Y values are in the range.
It is conventional to use Y for the dependent
variable and X for the independent variable,
but this convention is not at all required.
Now, to show this mapping we use a "graph."
Here's an example:
The arrows show how the letters are mapped
into numbers and represent the function f.
Using functional notation
we can say:
f(A) = 4
(read: f of A is 4)
likewise f(C) = 1
( f of C is 1 )
In general we would say Y = f(X)
With functions that relate 2 variables, both
of which are numeric, we can use coordinate
axis to show the graph of the function.
Next is an example of such a graph:
(This table specifies the function by
associating the independent variable to the
dependent variable.)
|
First Coordinate, the independent variable |
Second Coordinate,
the dependent variable |
| 0 |
0 |
| 1 |
3 |
| 2 |
4 |
| 3 |
2 |
| 4 |
5 |
| 5 |
1 |
| 6 |
8 |
| 7 |
6 |
| 8 |
7 |
| 9 |
10 |
| 10 |
9 |
The graph of these numeric quantities above
has two scales, one horizontal and one
vertical. These two scales represent the
the coordinate axis of the plot. (These
two scales usually intersect at the coordinate
0 on both, but this does not have to be the
case in general.) The
coordinates of the graph are the rows in the
table above; the first column contains points
along the horizontal and the second column
along the vertical. We specify a pair of
coordinates using parenthesis and a comma,
like so: (0,0), (1,3), (2, 4) ... (10,9).
So, for example, the point that represents
the coordinate pair (2,4) is found by first
finding 2 along the horizontal axis, then
moving vertically to 4 marked by the vertical
axis. Another way to visualize this
point is to find the intersection of the
vertical line that passes through 2 on the
horizontal axis and the horizontal line that
passes through 4 on the vertical axis.
Since we are dealing with 2 variables we
are talking about a two dimensional coordinate
system. Three variables would be shown
with a three dimensional coordinate system.
Higher dimensions are treated mathematically,
not graphically.
Here's another example of a graph specified
by either y = x2 or f(x)
= x2
Note that in this case, unlike the previous
graph where individual points were shown, a
curve traces the function along all points in
the domain from x = -3 to x = +3.
One final note, to connect the
points in the second graph is erroneous since the
nature of the function for values not in the
domain of that function is unknown.
Perhaps this function is a set of straight
lines, curves, or just undefined between the
known x coordinates. In spite of
this fact, interconnecting lines are sometimes
shown to show the overall shape of that graph.
More commonly, vertical bars centered at
each point, half the distance between successive
points, are used to create the so called bar chart.
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