Definition of a Relation
and a Function
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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 domainThe second elements in the ordered pairs (the y-values), 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 x-element has only ONE y-element associated with it.

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.

Vertical line test:

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