Section 1  Functions vs. Relations (Vertical Line Test)
The first section deals with the difference between relations and functions. It explains how to represent relations and functions using both mapping diagrams and graphs. It also explains how to determine whether or not a relation is a function, given a representation of that relation.
Relation  A set of inputs and outputs, often written as ordered pairs (input, output).
A "relation" is just a relationship between sets of information. Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is an example of a relation. In relations and functions, the pairs of names and heights are "ordered", which means one comes first and the other comes second. The set of all the starting points is called "the domain" and the set of all the ending points is called "the range." The domain is what you start with; the range is what you end up with. The domain is the x's; the range is the y's. (I'll explain more on the subject of determining domains and ranges later.) 
Function  A relation in which each input has only one output. Often denoted f(x).
A function is a "wellbehaved" relation. Think of when you are buying gas, and think of the price you would pay for a certain number of gallons. The pairing of gallons of gas and price is a functions. Again in relations and functions, the pairs of gallons and the price for that number of gallons are "ordered," which means one comes first and the other comes second. But this relation is a function because each input has only has one output. The different number of gallons of unleaded gallons you fill up, gives you one price you would have to pay. 
A function is a special relationship where each input has a single output. For every input, there is only one unique output.
(Warning: While all functions are relations, since they pair information, not all relations are functions. Functions are simply a subclassification of relations.)
When we say that a function is "a wellbehaved relation," we mean that, given a starting point, we know exactly where to go; given an x, we get only and exactly one y.
For a relation to be a function, there must be only and exactly one y that corresponds to a given x. Here are some pictures of this:
When we say that a function is "a wellbehaved relation," we mean that, given a starting point, we know exactly where to go; given an x, we get only and exactly one y.
For a relation to be a function, there must be only and exactly one y that corresponds to a given x. Here are some pictures of this:
In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xyplane more than once then for one value of x the curve has more than one value of y, and so, the curve does not represent a function.
Summary:

"Is it a function?"  Quick answer without the graph
Think of all the graphing that you've done so far. The simplest method is to solve for "y =", make a TChart, pick some values for x, solve for the corresponding values of y, plot your points, and connect the dots, yadda, yadda, yadda. Not only is this useful for graphing, but this methodology gives yet another way of identifying functions: If you can solve for "y =", then it's a function. In other words, if you can enter it into your graphing calculator, then it's a function.
The calculator can only handle functions. For example, 2y + 3x = 6 is a function, because you can solve for y:
The calculator can only handle functions. For example, 2y + 3x = 6 is a function, because you can solve for y:
2y + 3x = 6
2y = –3x + 6
y = (–3/2)x + 3
2y = –3x + 6
y = (–3/2)x + 3
Homework