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# Determining Whether a Relation is a Function

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Understanding relations (defined as a set of inputs and corresponding outputs) is an important step to learning what makes a function. A function is a specific relation, and determining whether a relation is a function is a skill necessary for knowing what we can graph. Determining whether a relation is a function involves making sure that for every input there is only one output.

One of the things you guys learn in your math classes is that there's ton of vocabulary to keep straight in your head. So we're going to look at a few vocabulary words before we start looking at numbers.

First is the domain. The domain is the set of all X. Sometimes called input values. The range is the set of all Y or output values. That will make more sense as we start looking at like actual numbers. In order for a relation to be called a function, each X value must have exactly one Y value. Function is a really important word in math class, and we're going to practice that more and more.

So let's start looking at some actual numbers where this will make more sense. Before we do that, keep in mind each X has to have exactly one Y value. You can't have two Y values and you can't have no Y values. So that's something to keep in mind. It has to have exactly one Y value. So let's look at a couple of examples.

A lot of times you guys see math information organized in a table. Here I have my X numbers, 8, 9, 10 and 13. That's kind of weird. Don't be freaked out. Sometimes math numbers are consecutive, like 8, 9, 10. Sometimes there's a wildcard thrown in there like 13. Don't worry, it's okay. Still going to work.

That's my domain, 8, 9, 10, 13. My range is my Y numbers. Negative 1, negative 3, 5. Let me show you another way you might see this written. And that's using ordered pairs. And you guys have seen ordered pairs before when you first learn how to graph. It's like a point, right? So I could write 8 comma negative 1 and that would represent X equals 8, Y is negative 1. I'm going to go through and write all of these in the point notation.

And then one thing that's kind of weird is we use these little curly brackets. This is called set notation. And you'll get into this more in your future.

And then the last way you might see this is through what's called a map or a mapping. And I'm going to draw these little bubbles. 1 is going to represent my Xs or my domain. The other is going to represent my Ys or my range.

So I have the X numbers 8, 9, 10 and 13. I have the Y numbers negative 1, negative 3 and 5. Notice that I didn't write negative 1 twice. Even though negative 1 shows up twice in the table and shows up twice in the ordered pairs, it only shows up once in the mapping.

And the thing I like about this is it's kind of easy to see what number goes with which when you draw arrows. Like 8 arrow to negative 1. 9 goes to negative 3. 10's also going to go to negative 1 so I'm going to have that double arrow coming into negative 1 and then 13 is matched with 5. Okay.

So those are all different ways to represent the same relationship. Now I want to think about is whether or not this is a function. Keep in mind in order to be a function each X value must have exactly one Y value. So go through and look. Does each X have a Y? Yes. You want to make sure that no X number shows up twice.

In our case this is a function, because each X number has exactly one Y number, but watch this. If I were to stick in 8 again with something that's not negative 1, like 8 and, I don't know, 15 or something, this would not be a function. Because now 8 has two different Y values. Let me get rid of that because nonfunctions kind of freak me out. Okay.

Important thing to keep me in mind is that each X number has to have exactly one Y number. And that will be called set of X is the domain and we call the set of Yes the range.

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