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# Relations and functions

## Video transcript

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 could view them as the set of numbers over which that relation is defined and then you have a set of numbers that can be you can view it 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 one with the number two in the range so in this type of notation you would say that the relation has 1 comma 2 1 comma 2 in its 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 to 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 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 I want to say negative 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 the net in a few seconds I'll show you what 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 one I say hey definitely maps to two you give me - definitely Maps tattoo as well you give me three it's definitely associated with negative seven 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 one is associated with two so let's build the set of ordered pairs so one is associated with two let's say that two is associated with let's say that two is associated with negative three so you'd have two negative three over there and let's say on top of that we also associate we also associate one with the number four so we also create an association with one with the number four so we have the ordered pair 1 comma 4 now this is a relationship we have it's defined for certain if this was the whole relationship then the entire domain is just the numbers 1 2 actually just the numbers 1 & 2 it's definitely a relation but this is no longer function and the reason why it's no longer a function is if you tell me okay I'm giving you 1 in the domain what member of the range is 1 associated with it 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 if you give me 1 I know I'm giving you a 2 you give me 2 I know I'm giving you 2 now that on 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 the possible you could 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 three or you could have a positive three 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 two four five two four five six six and eight two four five six and eight I could have drawn this with a big you know a 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 I'll now let's draw the actual associations so negative three is associated with two or it's a map to two so negative three maps to two 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 show you that first ordered pair let me that first ordered pair I should I don't want to get the 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 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 zero is associated 5 more 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 and it's now this this ordered pair saying it's also mapped it's also mapped to 6 and then finally 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 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 your output 4 or output six and because there's this confusion this is not a function you have 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