College Algebra: Piecewise-Defined Function Graphs

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Let's begin by graphing some of these really exotic functions that are referred to as piecewise functions. Those are functions that are actually defined differently on different parts of the interval of the x-axis. That is to say, you think of the x-axis as being over here, maybe the function is going to be a linear thing over here for a while, but then it just stops being linear and over here it's maybe a parabola kind of thing. Then it stops being a parabola and it's something else over here. Well, these sort of jigsaw puzzle like graphs can be graphed by just graphing each little window separately and putting the pieces together.
Let's try an example. For example, here's a function, f(x) equals--and now it looks pretty complicated and threatening, but let me remind you of how to read these things. This says, f(x) is a function and it equals one of two things, depending upon what x is. If x is bigger than, or equal to 0, then it equals 2x + 1. But if x were to be negative, less than 0, then it just equals x. So, this is a function that's actually made up of two functions that have been spliced together, right along the line x = 0. Then how would you graph this? Well, the way I would graph this, is I would graph each interval separately. So, for example, what I see here is the following.
Let's graph in green. If you take a look at this, here are my axes. So, the first thing I notice is that if x is bigger than or equal to 0--now what region is that? Well, x is 0 is here and bigger than 0 is here. So, it's this region, it's this half of the world. This half of the world--I want to graph that. So, let's graph that. Well, that's a line that has y intercept 1. So, I go up to 1 and it has slope, 2 over 1. So, that means I go 1 over and 2 up. There is the line, but I want you to remember that I'm not, I'm not going to graph the line all the way through, but only for when x is greater than or equal to 0. So, I'm going to start here and go out. It looks like this. I stop right there.
Now, what happens when x is less than 0. That's this region here. Well, then I'm going to graph a completely different function. I'm going to graph the function f(x) = x. Well, that has a y intercept 0, because it's x + 0 and the slope is 1. That means that I go 1 over and 1 up. What does that mean? Well, that means I go 1 over and 1 up, but wait a minute, 1 over and 1 up. 1 over, and 1 up, so it's here. But remember, I'm only graphing it down this way. If I put a point here and go 1 over and 1 up, I see the line, but I have to remember that that line, in fact, is only going to be this part of the line. So, it's not, let me put the whole line in there, so you can see it. That whole line would look like this. But I only want the part here. So, I'm just going to put in that part. In fact, there it is. This part is steeper, because it has slope 2. This is less steep, it has slope 1.
What happens, actually, at 0? Well, at 0 it's defined to be this. So, the way that I'll denote that is by putting an open circle right here, and then have my function live here. So, this comes up and then there's a jump and then it's steeper and goes up here. So, there's an example of a piecewise defined function and its graph. Let's try another one.
This one, f(x) equals one of two things. It either equals x + 1 if x is less than or equal to 3, or it equals the constant number 4 if x is bigger than 3. How would I graph this? Well, let's think about this for a second. Put in some axes here and what do I see? Well, the important point here is x being around 3. So, let's mark that point. Now what happens? If x is bigger than 3 the value of the function is a height of 4. So, it's just 4 and it's four no matter what x is. Right? If you plug in x = 5, it's bigger than 3 so the value is 4. You plug in 20, it's bigger than 3, so the value is 4. So, no matter where I am, the value is 4. It's constant. But actually, it's not defined right there to be 4 at 3. So, I put an open circle because the inequality is strict.
What do I do at 3? When x = 3, I go up to this thing. There I see what? I see 3 + 1, oh, that is 4. So, I'm allowed to color that in because it's part of this graph. What is this graph? This is a graph that has slope 1 and y intercept 1. So, I go up 1 and slope 1 goes right to there and then comes down. So, this piecewise defined function actually has a graph that looks like this thing and then it sort of sharp turns to the right and I've got this constant function. The constant functions given by this and this is a line function that has slope 1 and intercept 1. So, again, you can see how there's a break in the action usually where the thing splits.
Let's try one last one. This one's a real exotic one. This one's a triple action one. You're equals one of three things, depending where you are. If you're strictly smaller than -4, then you're 2x + 1. Where did I get that from? It's 2 + x. If you're between -4 and 5, then you equal -x. If you're bigger than 5, then you equal 3x. So, what in the world does this look like? Now, I have to graph three things separately and try to put them all on one axis. Let's see if we can do this.
What I'm going to do here is the following. The important thing here, by the way, is just to get the scale right. It looks something like this, I think. Let's see, -4--let's hope for the best. So, the important points where things are going to change are going to be when x is around -4 and also when x is 5. So, let me just put those points down there, just to make sure I'm going to what out for those. There's -4. There's 5. So, I know things are going to happen between there and I've got to be careful.
What happens if I'm less than -4? If I'm less than -4, then I see I'm going to be graphing 2 + x. What is 2 + x? Well, 2 + x is the line that has intercept 2, y intercept 2. Don't be fooled because the 2 is in front. You might say, "Oh, the slope is 2." No, no, no, slope is the coefficient in front of the x. So, that's just 1, but the y intercept is 2. So this has slope 1, so the line looks like this. That's the line. I don't draw it like that because notice that line only kicks in from this region. So, it only starts right here, at -4. You see? Because I don't look at it except if x is less than 4. That means from here on out. So, I'm just going to take that part of the function and ignore everything else. Let me draw that in and let's see what that looks like.
That's just that piece of the function from -4 and to the left. I threw this stuff away; I just threw that stuff away, because this is the line only in that region and nowhere else. What happens now between -4 and 5? Well, between -4 and 5, I'm -x and -x, I remind you, just looks like this. But I only want that between this region. So, what I want to do, is--well, I'm going to start up here at -4. If I plug in -4 in for x, I see -(-4), which is +4. So, this is a height of 4. Then I go through here, all the way to 5. Where if I plug in 5, I see -5, and I want all the points in between. So, it looks like this. There's that part. That's just that line y = -x, but only in between here and here. It's like a window and I've covered everything else up.
Now what happens when x is bigger than 5? If x is bigger than 5, then I'm looking a 3x. What's 3x? That goes through the origin and it has slope 3 over 1. That means--but I have to start it here at 5. That's going to be way up here. In fact, you know it's going to be off the screen here. I can't even show you this, but I'll just show over there, really carefully. Look what I'm going to do. I'll draw it even though you can't see it, but watch over there. I'm going to put the point there at 5. It goes up, way up around 15 and then its slope is going to be really big. It's 3 over 1. So, you can see it looks like that.
You can see this thing has these different pieces. What dots should I color in? Well, wherever there's a equal sign. So, what I'm looking at is the following. I'm looking at -4, so I color that in, and I'm looking at the 5, so I color that in. That function is where these endpoints are defined and these are open circles. Really hard problem, but we resolved it. I'll see you soon.
Relations and Functions
Graphing Functions
Graphing Piecewise Defined Functions Page [2 of 2]