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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.
In the general sense mapping values has no constraints, that is, all, some, or none
are mapped. In our study from this point through calculus we will require
this mapping to map one value in the domain to one or more values in the range.
We will not allow a single value in the domain to map to more than one value in the range.
Nor will we require the entire domain to be mapped into the range.
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)
Examples
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(1)
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Each value in the domain is mapped to one value in the range,
so this is a function.
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(2)
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1 and 2 both map to -1. Yet they each map to one number only.
This fits the definition of a function as far as we are concerned.
Each value in the domain is mapped to one value in the range,
but not all values (0 and 2) in the range are used. This is still ok.
This is a function.
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(3)
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In this example 3 maps to both 0 and 4. This mapping violates our
definition of a function, therefore this is not a function.
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(4)
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Not all values are mapped, but those that are are only mapped to a single
value. This is a function.
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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:
The following table specifies the function by
associating the independent variable to the
dependent variable. The range is specified in the left column and the domain
is specified in the right column. Numbers not listed in this table are not part of the domain
or range.
|
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. 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 since it is difficult to graphically
represent functions beyond 4 dimensions.
One final note about this function,
to arbitrarily connect the
points in the first 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, we can model this function with a
curve to help understand the trend of this function and
even to make an estimation beyond these points.
A common techinque is to draw vertical bars centered at
each point, half the distance between successive
points, to create the so called bar chart.
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.
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