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Relation:
A relation is simply a
set of ordered pairs. |
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A relation can be any
set of ordered pairs.
No special rules need apply.
The following is an example of a relation:
relation:
{(1,2), (2, 4), (3, 5), (2, 6),
(1, -3)}
The graph at
the right shows that a vertical line may
intersect more than one point in a relation.
The graph of this orange
arrow is also a relation. |
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If we impose the
following rule on a relation, it becomes a function.
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Function:
A function is a set of
ordered pairs in which each x-element
has only ONE y-element associated
with it. |
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The relation shown above is NOT a
function because the x-element 2 is paired with a
y-element of 4 and also a y-element of 6.
Similarly, the x-element of 1 is paired with the
y-elements of 2 and -3.
The relation above can be altered to
become a function by removing the ordered pairs where
the x-coordinate is used twice.
function:
{(1,2), (2,4), (3,5)}
The graph at
the right shows that a vertical line intersects
only ONE point in a function.
This is called the vertical line test for
functions. |
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In
a function, once the x-coordinate
finds its partner, it never pairs up
with any other values. |
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A function may not have
two y-values assigned to the same x-value,
such as {(2,4), (2,6)}.
A function may have two x-values
assigned to the same y-value, such as {(2,4),
(3,4)}.
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Let's take one last look at relations and
functions.
Consider the following relations. Are they
also functions?
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Situation 1 |
Chico Lynn Paul |
Consider the relation described as:
(x, y) = (student's name,
shirt color)
This relation
consists of
{(Chico, gray), (Lynn,
gray), (Paul, gray)}
This relation is
also a function!
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Now, let's reverse the situation:
(x, y) = (shirt color,
student's name)
This relation consists of
{(gray, Chico), (gray,
Lynn), (gray, Paul)}
This relation is
NOT a function!If you
are told that the student
wearing the gray shirt wants to
ask you a question, how will
you know which student to
approach?
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Situation 2 |
Chico
Lynn Paul |
Consider the relation described as:
(x, y) = (student's name,
shirt color)
This relation
consists of
{(Chico, gray), (Lynn,
green), (Paul, red)}
This relation is
also a function!
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Now, let's reverse the situation:
(x, y) = (shirt color,
student's name)
This relation consists of
{(gray, Chico), (green,
Lynn), (red, Paul)}
This relation is
also a function!In
this situation, if you are told
that the student wearing the
gray shirt wants to ask you a
question, you will know that
the student is Chico.
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Traditionally,
functions are referred to by the notation f (x),
but f need not be the only letter used in
function names. Remember that f (x)
is telling you that the result will be "a function
of x", or is dependent upon x.
The statements y = x2
and f (x) = x2 are
basically the same.
You may even see statements such as f (x)
= y = x2 .
Example: A function
is represented by f (x) = 2x + 5.
Find f (3).
To find f (3), replace the x-value with 3.
f (3) = 2(3)
+ 5 = 11.
The answer, 11, is called the
image of
3 under f (x).
Note: The
f (x) notation can be thought of as
another way of representing the y-value,
especially when graphing. The y-axis may
even be labeled as the f (x) axis.
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