 This video is going to talk about piecewise defined functions and the thing that you need to remember about piecewise defined functions is that it's a bunch of different graphs but it's not the whole graph, it's just pieces. We're going to look at a certain part of the domains. We only want little pieces in our graph, a bunch of little pieces. So the domain is going to be what defines those little pieces. We also have to talk about continuous and not continuous when we talk about piecewise defined functions. A continuous graph would be like when you draw a line, your pencil stays on the graph for the whole line from arrow to arrow or even from endpoint to endpoint if we only have a section of it. But when you have pieces sometimes you're going to have to lift your pencil to get to another part of the graph so that gives you a not continuous graph. Let's look at our first example here. It says that we have this x plus one but it's only between negative two and two in the x's. In fact it doesn't even include two over here. And then we have the x and that's going to be from two including two up to four and including four. Again that's what this part A is telling you but we want to graph that one. So let's graph and say that if I had x equal negative two then it would be negative two plus one. And negative two plus one is going to be equal to negative one. So negative two, negative one should be on my graph and if I let x equal zero so I'd have zero plus one which is going to be one. So zero one should be on my graph. And then we have the two is our other endpoint. So we need to check that one out for sure. Two plus one is going to be three. So two three, one, two, three but it doesn't include that one because it doesn't have an equal in the inequality. So my graph only goes from this negative two one up to that two three and not including it. And then the next part of the graph, I'll do that over here and we'll do that in a different color. We have just x. So when x equal two it's going to be two or two would be on our graph down here. And then we go from three and that's just going to give us x. So three three is going to be on our graph and then four four would be on our graph and it's included because the four has an inequality that includes it. So I have a line between these two points and they're both filled in and we can see now that we have this graph. So part B says what's the domain? Well it starts at negative two. There's nothing over here beyond negative two to the left. So it starts at negative two and it includes it and it goes all the way and here at two it's in the red it's not included but it is included. There is a y value at x equal two on the blue line. So we can say that it goes from negative two all the way over to four. There is some value on one of the graph or the other that has an x value for every number in that interval. We want to talk about being continuous or not continuous and here's where you see that we have that not continuous thing happening. I had to lift my pencil from up here to drop down to the blue. So it's not continuous and then we've drawn our function. Let's graph this. So we have two x plus three when x is less than zero so we just need to get a couple points here. So if we try f of say negative one we'd have two times negative one plus three and negative two plus three would be equal to one. So negative one one would be on our graph which would be this point right here. If we try to say like negative three just to get a second point it'd be again putting a plug in it in you could probably do this in your head but just in case negative six plus three is going to be negative three. So negative three negative three would be on our graph. It's going to be anything less than zero so it doesn't include the zero. If we put zero in here two times zero would be zero plus three. So zero three would be on our graph but it's going to be an open circle because it doesn't include that point but then it goes through all the rest of these points and put an arrow on it because it'll be all the rest of them forever. Now the second piece x squared plus one we're only going between zero and almost two. So we do f of zero and that's just going to be one. Zero squared plus one would be one. So zero one would be on this graph and it is included. And then we could try f of one and if we do one it'd be one squared plus one or just two. So one two is on our graph and then two f of two would give us two squared which is four plus one is going to be five so we have the point two five and it's included no it's not so it's an open circle. And it starts from here at zero one and then goes up to that point but not including it. And notice it's just a little piece of that graph. Now we go to the next piece and it says five. Our graph is going to be equal to five whenever x is greater than two. Again it doesn't include two. It starts at two and goes forever at this value five as a constant. So our graph is going to look like this. Okay the arrow is telling me it's going to keep going forever. So now let's look at what we need to do with it. It says is the graph continuous? This is you ask yourself can you pick up your pencil? If you can pick up the pencil then no. If you can pick up your pencil then it's not continuous. So anytime there's gaps in your graph it's not going to be continuous. Now it asks me to evaluate several points. H of negative three. Negative three fits in the domain of x is less than zero. It does not fit in between zero and two and it doesn't go into the domain of x is greater than two. So this is going to be two times negative three plus three only. And we already did that one. We found out that that was negative three. If I do H of two it doesn't fit into less than zero. It doesn't fit into between zero and two because it doesn't include two here. And it also doesn't include two here. So two is not included anywhere so it's not let's say not in domain. And then we have H of three. And H of three would go into this part of the graph that where x is greater than two. And that just tells us that it's going to be equal to five. Now it asks us to find the domain and range. For the domain it's going to be from negative infinity because it goes this way forever. And then it goes across here and all those points are included. Zero is included. Keep going on and on forever. And it's going to be all except for this two because that's the only place where there was no value. When x was two there was no y value. Over here when x was zero I had a value here but not on the first piece. But as long as there's a value somewhere when x is zero then we can include it. So we would say that it goes from negative infinity up to two but not including two. And then union two to infinity. So that includes everything but that two. Again here it works because we do have a point at x equals zero. It's just on the second piece. Now the range. The range goes from negative infinity because it's going down forever. And it goes up just to this five. It doesn't go any higher than five because the last piece just says it's equal to five. So let's do another one. Graph the function and this time we're going between negative three and two. So we look at x squared if it were negative three squared have nine be this point right here. And it includes it. If we did negative two it would be four. And if we did one it would be one squared is one. If we did zero zero squared is zero. And if we did one it would be one and two squared is four. But it's an open circle at two four because it does not include the two. So that's a quadratic and you can see that it's a quadratic. You should have known that it was going to be a quadratic just by looking at this x squared. But you might not have known exactly how it was shaped. So it's good. Lots of times you can just do endpoints. But on quadratics you need to get several points so you can see how it curves. Now when x is greater than or equal to two it's going to be four. Well that means that now for the second piece in fact let me do it in a different color. The second piece I'm going to get to include that point and it's all the rest of them. And they will all have fours as their y value. So let's go and then answer our questions. Is a graph continuous? This time it is continuous because this point was open for the first piece but it got closed from the second piece. So two has a value. When x is two there is a y value for that one. So it is a continuous graph. H of negative three. We did that one up here. That's equal to nine. But notice it doesn't fit into the second piece because it's not greater than or equal to two. H of two it's greater than or equal to two. So I can only look at this bottom piece. I don't want to look at the top piece because it's not equal to two there. X is only equal to two in the bottom part so we know that that one's four. And then we have H of three. Three doesn't fit in the domain of negative three to two. But if it does fit into x it's greater than two and so we again have a four. We'd evaluate it at the second piece. Our domain starts here at negative three. I didn't put an arrow on here because it was less than or equal to negative three. So it starts at negative three and it goes forever. This arrowhead tells me that I'm going to have all the rest of the x's so it goes to infinity. And I need to bracket here at three because the absolute value here said including negative three. My range. It only went as high as nine. There's no arrowhead again here. So it only went as high as nine and it only went as low as zero. So the smallest value first it included zero and it went up to nine and included that one too.