Definition of a Relation and a Function
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.
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.
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.
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.
