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Graphs and Functions
Relations and Determinig Whether a relation is a Function

A relation is a set of ordered pairs.

There are several ways to write a relation:

You can list a set of ordered pairs:

{(8,-1), (9,-3), (10,-1), (13,5)}

You can draw a mapping diagram as shown below: The points in the mapping diagram are the same points that were listed in the set of ordered pairs. The first oval contains the x-values and the arrow points to the appropiate y-value.

You can also write a relation in a table:

 x y 8 -1 9 -3 10 -1 13 5

You can display a relation on a graph: or finally you can write a relation as an equation: y = x2

A function is a relation where for every x there is exactly one value for y.

To classify relations as functions when you are given a list, a mapping diagram, or a table, check the x-values. If there are two x-values that are the same but have different y-values, then the relation is not a function. If all the x-values are different and have different y-values, then the relation is a fuction. This is a function! This is NOT a function!

To classify relations as functions when you are given a graph, you can use the Horizontal Line Test: If a vertical line intersects more than one point on the graph, then the relation is not a function. If a vertical line intersects the graph in only one point, then the relation is a function. This is NOT a function!

To classify a relation as a function when you are given an equation, then graph the equation and look at the graph.