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 a 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 that 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 or it could point to Z or maybe it could point to e or whatever else this would not be a function because over here so this right over here not a function 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 anymore 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 range is or I should say 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 out or Y is going to be equal to 3 so one way to write that mapping is you could say you could say X when you input it let me write it this way negative 1 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 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 2 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 to 4x 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 it 1 so it's not defined there so 1 isn't part of the domain 2 is it tells us when X is - then why is going to be equal to negative two so it Maps it or associates it with negative two that doesn't seem too troublesome just yet now let's look over here our function is also defined at X is equal to three it associates three our function associates or Maps three to the value y is equal to two Y is equal to two that seems pretty straightforward and then we get to X is equal to four where it seems like this thing that could be a function it is the it is somewhat defined it does try to associate four with things but what's interesting here is it tries to associate four 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 four with five do we associate with five or do we associate it with negative one so this thing right over here is actually a relation you can have 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 the function so once again because of this this is not not a function it's not clear that when you input X when you input four into it should you output five or should you output negative one 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 okay when X is four 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 that value in the domain there's two or more outputs for the input four 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