>> A function is just a special kind of relation and here is a definition. A function is a relation in which every element in the domain, every value of X, if we're thinking of X's and Y's, corresponds with exactly one element in the range. So a function is just a set of ordered pairs in which for every value that X takes on, there is exactly one value that Y takes on. Alright. Sort of using primitive graphs, the situation looks like this. Now in our domain in range values here in the previous example, we had 1 associated with 2, 2 associated with 4, 3 associated with 6. Now do we need the definition of function for this particular relation? Well let's see. Is it true that for every element in the domain there is exactly one element in the range? Well does this guy have one partner over here? Yup! Does this guy have one partner over here? Yup! Does this have one partner over here? Yes. So this is function type of relationship. How about this one? Now notice the correspondence here. We have 7 corresponding with 4, 8 corresponding with 5, 9 corresponding with 5. Do we meet the requirements of function? It turns out that we do here because every element in the domain corresponds with only one element in the range. Now these two guys happen to be corresponding with the same element in the range but that's okay. You see, we have only one value in the range associated with 8, only one value in the range associated with 9. And by the way, there could be other items in the range. The range could include 1 and 10 that have no correspondence with items in the domain. It's okay. If we had other items in the domain though, we would need to have partners in the range corresponding with them. That's the idea of exactly one. That means one and only one. So we have to have a partner and only one partner, you see, in the range for us to have a function relationship. Alright. Here's an example of a non-function relationship. We have 7 and 8 in our domain and the range consists of these elements and the correspondences are 8 corresponds with 6, 7 corresponds with 4, 7 also corresponds with 5. And this situation is what causes this to be not a function relationship because 7 corresponds with two items in the range. Now we're gonna look at a lot of different ways to describe relationships. You know, we have--we can describe relationships in mathematics between variables with ordered pairs. We can describe relationships sort of verbally. We can describe relationships graphically. We can describe relationships with an equation. And we're gonna take some time here and talk about function versus non-function in all of those various situations. Here we're looking at ordered pairs. Now another way to kinda look at the situation and it's kinda paraphrasing what we have here, is this, that every value that X can assume is associated with only one Y-value. You see, it's the same exact idea, we just don't like to confine ourselves to X's and Y's.