Relation:
A relation is simply a
set of ordered pairs. 
The first elements in the ordered pairs (the
xvalues), form the domain. The second
elements in the ordered pairs (the yvalues), form the
range. Only the elements "used" by the
relation constitute the range.

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).
•
The domain is the set {1, 3, 5, 9}.
•
The range is the set {2, 7, 8}.
(Notice that 3, 5 and 6 are not part of the range.)
•
The range is the dependent
variable. 

The following are examples of relations.
Notice that a vertical line may intersect a
relation in more than one location.

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.

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.
Function:
A function is a set of
ordered pairs in which each xelement
has only ONE yelement associated
with it. 
The relations shown above
are NOT
functions because certain xelements are paired with
more than one unique
yelement.
The first relation
shown above can be altered to
become a function by removing the ordered pairs where
the xcoordinate 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. 

Vertical line test: 
each vertical line
drawn through the graph will
intersect a
function in only one
location. 

