Recognizing functions from graph

Determine whether the points on this graph represent a function. Now, just as a refresher, a function is really just an association between members of a set that we call the domain and members of the set that we call a range. So if I take any member of the domain, let's call that x, and I give it to the function, the function should tell me what member of my range is associated with it. So it should point to some other value. This is a function. It would not be a function if it says, well, it could point to y. Or it could point to z. Or maybe it could point to e or whatever else. This would not be a function. So this right over here not a function, because it's not clear if you input x what member of the range you're going to get. In order for it to be a function, it has to be very clear. For any input into the function, you have to be very clear that you're only going to get one output. Now, with that out of the way, let's think about this function that is defined graphically. So the domains, the valid inputs, are the x values where this function is defined. So for example, it tells us if x is equal to negative 1-- if we assume that this over here is the x-axis and this is the y-axis-- it tells us, when x is equal to negative 1, we should output. Or y is going to be equal to 3. So one way to write that mapping is you could say, if you take negative 1 and you input it into our function-- I'll put a little f box right over there-- you will get the number 3. This is our x. And this is our y. So that seems reasonable. Negative 1 very clear that you get to 3. Let's see what happens when we go over here. If you put 2 into the function, when x is 2, y is negative 2. Once again, when x is 2 the function associates 2 for x, which is a member of the domain. It's defined for 2. It's not defined for 1. We don't know what our function is equal to at 1. So it's not defined there. So 1 isn't part of the domain. 2 is. It tells us when x is 2, then y is going to be equal to negative 2. So it maps it or associates it with negative 2. That doesn't seem too troublesome just yet. Now, let's look over here. Our function is also defined at x is equal to 3. Our function associates or maps 3 to the value y is equal to 2. That seems pretty straightforward. And then we get to x is equal to 4, where it seems like this thing that could be a function is somewhat defined. It does try to associate 4 with things. But what's interesting here is it tries to associate 4 with two different things. All of a sudden in this thing that we think might have been a function, but it looks like it might not be, we don't know. Do we associate 4 with 5? Or do we associate it with negative 1? So this thing right over here is actually a relation. You can have one member of the domain being related to multiple members of the range. But if you do have that, then you're not dealing with a function. So once again, because of this, this is not a function. It's not clear that when you input 4 into it, should you output 5? Or should you output negative 1? And sometimes there's something called the vertical line test that tells you whether something is a function. When it's graphically defined like this, you literally say, OK, when x is 4, if I draw a vertical line, do I intersect the function at two places or more? It could be two or more places. And if you do, that means that there's two or more values that are related to that value in the domain. There's two or more outputs for the input 4. And if there are two or more outputs for that one input, then you're not dealing with a function. You're just dealing with a relation. A function is a special case of a relation. Or you could view it as a well-behaved relation.