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 xvalues and the arrow points to the appropiate yvalue. 
You can also write a relation in a table:
You can display a relation on a graph:
or finally you can write a relation as an equation: y = x^{2}
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 xvalues. If there are two xvalues that are the same but have different yvalues, then the relation is not a function. If all the xvalues are different and have different yvalues, 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.