Topic A: Functions
Is the relation given by the set of ordered pairs shown below a function? So before we even attempt to do this problem, right here, let's just remind ourselves what a relation is and what type of relations can be functions. So in a relation, you have a set of numbers that you can kind of view as the input into the relation. We call that the domain. You can view them as the set of numbers over which that relation is defined. And then you have a set of numbers that you can view as the output of the relation, or what the numbers that can be associated with anything in domain, and we call that the range. And it's a fairly straightforward idea. So for example, let's say that the number 1 is in the domain, and that we associate the number 1 with the number 2 in the range. So in this type of notation, you would say that the relation has 1 comma 2 in its set of ordered pairs. These are two ways of saying the same thing. Now the relation can also say, hey, maybe if I have 2, maybe that is associated with 2 as well. So 2 is also associated with the number 2. And so notice, I'm just building a bunch of associations. I've visually drawn them over here. Here I'm just doing them as ordered pairs. We could say that we have the number 3. 3 is in our domain. Our relation is defined for number 3, and 3 is associated with, let's say, negative 7. So this is 3 and negative 7. Now this type of relation right over here, where if you give me any member of the domain, and I'm able to tell you exactly which member of the range is associated with it, this is also referred to as a function. And in a few seconds, I'll show you a relation that is not a function. Because over here, you pick any member of the domain, and the function really is just a relation. It's really just an association, sometimes called a mapping between members of the domain and particular members of the range. So you give me any member of the domain, I'll tell you exactly which member of the range it maps to. You give me 1, I say, hey, it definitely maps it to 2. You give me 2, it definitely maps to 2 as well. You give me 3, it's definitely associated with negative 7 as well. So this relation is both a-- it's obviously a relation-- but it is also a function. Now to show you a relation that is not a function, imagine something like this. So once again, I'll draw a domain over here, and I do this big, fuzzy cloud-looking thing to show you that I'm not showing you all of the things in the domain. I'm just picking specific examples. And let's say that this big, fuzzy cloud-looking thing is the range. And let's say in this relation-- and I'll build it the same way that we built it over here-- let's say in this relation, 1 is associated with 2. So let's build the set of ordered pairs. So 1 is associated with 2. Let's say that 2 is associated with, let's say that 2 is associated with negative 3. So you'd have 2, negative 3 over there. And let's say on top of that, we also associate, we also associate 1 with the number 4. So we also created an association with 1 with the number 4. So we have the ordered pair 1 comma 4. Now this is a relationship. We have, it's defined for a certain-- if this was a whole relationship, then the entire domain is just the numbers 1, 2-- actually just the numbers 1 and 2. It's definitely a relation, but this is no longer a function. And the reason why it's no longer a function is, if you tell me, OK I'm giving you 1 in the domain, what member of the range is 1 associated with? Over here, you say, well I don't know, is 1 associated with 2, or is it associated with 4? It could be either one. So you don't have a clear association. If I give you 1 here, you're like, I don't know, do I hand you a 2 or 4? That's not what a function does. A function says, oh, if you give me a 1, I know I'm giving you a 2. If you give me 2, I know I'm giving you 2. Now with that out of the way, let's actually try to tackle the problem right over here. So let's think about its domain, and let's think about its range. So the domain here, the possible, you can view them as x values or inputs, into this thing that could be a function, that's definitely a relation, you could have a negative 3. You could have a negative 2. You could have a 0. You could have a, well, we already listed a negative 2, so that's right over there. Or you could have a positive 3. Those are the possible values that this relation is defined for, that you could input into this relation and figure out what it outputs. Now the range here, these are the possible outputs or the numbers that are associated with the numbers in the domain. The range includes 2, 4, 5, 2, 4, 5, 6, 6, and 8. 2, 4, 5, 6, and 8. I could have drawn this with a big cloud like this, and I could have done this with a cloud like this, but here we're showing the exact numbers in the domain and the range. And now let's draw the actual associations. So negative 3 is associated with 2, or it's mapped to 2. So negative 3 maps to 2 based on this ordered pair right over there. Then we have negative 2 is associated with 4. So negative 2 is associated with 4 based on this ordered pair right over there. Actually that first ordered pair, let me-- that first ordered pair, I don't want to get you confused. It should just be this ordered pair right over here. Negative 3 is associated with 2. Then we have negative 2-- we'll do that in a different color-- we have negative 2 is associated with 4. Negative 2 is associated with 4. We have 0 is associated with 5. 0 is associated with 5. Or sometimes people say, it's mapped to 5. We have negative 2 is mapped to 6. Now this is interesting. Negative 2 is already mapped to something. Now this ordered pair is saying it's also mapped to 6. And then finally-- I'll do this in a color that I haven't used yet, although I've used almost all of them-- we have 3 is mapped to 8. 3 is mapped to 8. So the question here, is this a function? And for it to be a function for any member of the domain, you have to know what it's going to map to. It can only map to one member of the range. So negative 3, if you put negative 3 as the input into the function, you know it's going to output 2. If you put negative 2 into the input of the function, all of a sudden you get confused. Do I output 4, or do I output 6? So you don't know if you output 4 or you output 6. And because there's this confusion, this is not a function. You have a member of the domain that maps to multiple members of the range. So this right over here is not a function, not a function.