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Relation:
A relation is simply a
set of ordered pairs. |
The first elements in the ordered pairs (the
x-values), form the domain. The second
elements in the ordered pairs (the y-values), form the
range. Only the elements "used" by the
relation constitute the range.
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This mapping shows a
relation from
set A into set B.
This relation consists of the ordered pairs
(1,2), (3,2), (5,7), and (9,8).
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The domain is the set {1, 3, 5, 9}.
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The range is the set {2, 7, 8}.
(Notice that 3, 5 and 6 are not part of the range.)
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The range is the dependent
variable. |
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The following are examples of relations.
Notice that a vertical line may intersect a
relation in more than one location.
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This set of 5
points is a relation.
{(1,2), (2, 4), (3, 5), (2, 6),
(1, -3)}
Notice that vertical lines may intersect
more than one point at a time.
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This parabola is
also a relation.
Notice that a vertical line can
intersect
this graph twice. |
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 relations shown above
are NOT
functions because certain x-elements are paired with
more than one unique
y-element.
The first relation
shown above can be altered to
become a function by removing the ordered pairs where
the x-coordinate is repeated. It will not
matter which "repeat" is removed.
function:
{(1,2), (2,4), (3,5)}
The graph at
the right shows that a vertical line now intersects
only ONE point in our new function. |
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Vertical line test: |
each vertical line
drawn through the graph will
intersect a
function in only one
location. |
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