>> Lecturer: Another family of functions would be those functions that are defined in pieces. They're called piecewise defined functions. And here is an example of one. I can't say that this is the most degenerate kind because it kind of doesn't fall into a category like that. But the function is defined in pieces and it's saying that, okay here is the definition of the function when X falls into a particular interval. And there is the definition of the function, here's how we graph it, when X falls into this other region, this other interval. So when X is less than or equal to 1, here's how we're going to define it. And here's how the graph is going to behave. And we're going to define it differently on this interval, and here's how the graph is going to behave. So when we make a table of values, we have to first decide what interval we're dealing with. If we're talking about negative, oh let's say negative 1 for X. Well, for an X of negative 1, we look over here. Which one of these do we use? Well, we come over here and we say, well this X that we've chosen is less than or equal to 1, so we're going to use this definition, you see? And so we plug X into this, so 2 times negative 1 would be negative 2. Negative 2 plus 3, oh that's 1. So the point negative 1, 1 is on the graph, you see? So that's the idea. So in all of this -- for all of the values of X that lie in this interval, we are using this definition, you see, to make the table of values. And if we jump over to a number like, oh let's say 3, then we're no longer in this interval for Xs, we're in this interval. So when X is greater than 1, and certainly X is greater than 1 if it's 3, we're going to use this definition. So replacing X with 3, we have negative 3 plus 4, oh that's 1. So 1, 2, 3 goes up to 1. And that's how we make the graph. Now notice that there's a little hole in the graph right here. And it's because this -- we use this definition when X is greater than 1. And that's all these values over here. But when X is equal to 1, we're using the definition up here and that's this point. So for every point to the right of 1, an X of 1, we define like this and we're on this graph. So we show the hole in the graph right here. We show that this point is not included and that one is. We certainly wouldn't want a solid point both here and here.