Common Core Algebra I.Unit 3.Lesson 7.The Domain and Range.by eMathInstruction

Hello and welcome to another common core algebra one lesson. By E math instruction. My name is Kirk Weiler, and today we're going to be doing unit three lesson 7. The domain and range of the function. Now, before we get on to this lesson, let me remind you that you can find a worksheet and a homework set that go with this video. By clicking on the video's description. As well, don't forget that at the top of our worksheets, we have our QR codes. Scan these with either your smartphone or your tablet, and it will bring you right to these videos. All right, let's begin this lesson. The domain and range of a function are very, very important. All right? But they are pieces of terminology that you're going to have to learn. Let's take a look at their technical definitions and try to understand them, and then we'll get into a variety of exercises that reinforce these definitions. Throughout many lessons during this course, domain and range will be coming up. So we're going to want to get used to the terms right now. So let's begin. The domain of a function is the set of all inputs for which the function rule can give an output. That's it. So if you've got an input that gives you an output, then that typically X value belongs in the domain. If you can't get an output, it doesn't belong in the domain. It's pretty much that simple. On the other hand, the range is the set of all outputs for which there is an input that results in them. Oh, that seems a little bit confusing. But basically, a number typically, it doesn't have to be a number. But something lies in the range, if it is an output to the function. In other words, if you can find an input that gives you that thing as an output, then it's part of the range. All right? So we're going to be working with both of these concepts throughout this lesson and throughout a variety of different lessons that show up in the course. So let's begin. Okay. So let's take a look at a function, which is defined by this graph. The first thing it asks us to do is something that we've done many, many times at this point, which is to evaluate a function based on its graph. I'd like you to pause the video right now and make sure you can do problem one letter a. All right. Let's take a look. Now remember, as always, all of these things inside the parentheses are X values. And the functions outputs are the Y values. So if I go over to X equals negative three, what I find is I find a Y value of one. If I go to X equals one, I find a Y value of negative three, just a coincidence there. And if I go to an X value of three. I find a Y value of one, two, three, four, 5. All right, simple enough. Now that question was really there to set us up for letter B can the function rule, given by the graph, give you a value when X is 5. If so, what is it? If not, why not? So see if you can answer letter B right now. Pause the video, see what you come up with. All right. Well, let's do it. Let's go over to one, two, three, four, 5, and. No. No. So you can't. No. And I would say it's because the graph. Doesn't. Extend that far. All right. Is X equal 5 in the domain of the function and the answer is absolutely not. No. And that's because no output can be found. And that is the real, the real basis for this piece of terminology, right? A value of X will be in the domain of the function if you can find a value of Y or an output for that value of X 5 can't line the domain simply speaking because there is no value of Y that goes with it. All right? So I'm going to clear out this text. Write down whatever you need to. All right, here we go. Continuing on with this problem. Letter D says give two other values of X that are not in the domain of the function. To see if you can figure that out, give me there's many other values of X, many, many, because we can obviously choose non integer values. We can choose values that have fractional parts or even irrational values. Although we may not be up to speed on irrational numbers. So pause the video now and come up with a couple more values of X that aren't in the domain of this function. That won't give you outputs. All right. Let's go through them. Well, as I said, many different choices. We already know X equals 5, but one, two, three, four. Four doesn't have anything. You could go with X equals 6, but let me go negative one, two, three, four, 5, 6. Hey, let me get tricky. Let me go X equals negative 5 and a half. That would be our friend right here. And again, nothing there, right? So both of those aren't in the domain of the function. Let's talk a little bit about the range of the function. Remember, the range are the Y values, right? They are the outputs to the function. Okay, I'm going to whoops. I'm going to clear some of this stuff out. So we get rid of it easier to see now. Move my little bar back over. Okay, well, you know, is Y equals zero in the domain? Well, here's a Y equals zero. Oops. Here's a Y equals zero, Y equals zero Y equals zero, so sure. That's in the range. How about Y equals three? One, two, three. Here's a Y equals three. That's in the range. How about a 6? One, two, three, four, 5, 6. All right, I'm gonna put an X there. No, right? How about 5 going back to blue? Negative 5. One, two, three, four, 5, negative 5s all along here. No. Do this. Let's go back to blue, negative one. Here's a negative one. And negative one and a negative one. So sure and how about four? One, two, three, four. Yeah, here's a four. All right, so all those values that have the blue checks by them, they're in the range. X's aren't. So now, let's write the domain. And the range. The domain are all the X values for which we have Y values. So the smallest X one, two, three, four, 5. The smallest X is negative 5. And we do get a Y value there. The largest X one, two, three, is three. So here's a good way to state it, but boy, this is a place where interval notation, if you're allowed to use it, works great. Because you can just talk about the first X value, the last X value, since they're both have Y values, they're included, so we use brackets. The range, this is a little bit trickier. Let's look for the smallest Y value. Here it is, right there, right? And that's one, two, three, four, negative four, and we're talking about Y's here. And the largest Y value is actually the one that comes up right here. And that's one, two, three, four, 5. So this is a good statement of the range or again an interval notation you gotta love it. Negative four to 5. With the brackets because we do hit the 5 and we do hit the negative four. All right? So a lot of students, if they retain anything about domain and range, the domain are the X values, the range are the Y values. It's not bad. It's definitely a first start. All right? We're going to want more than that eventually, but domain X range Y, okay? So I'm going to clear out this text write down anything you need to. All right, here goes the scrub. The clean sheet. Let's keep going. One of the things we haven't looked at very much are what are known as napping diagrams for functions. And they're really kind of cool because what functions ultimately do is they take numbers or just general inputs from the domain. And they map them to the range. All right? And it says given the function F of X equals X divided by two minus three. And the domain shown, fill in the range, write the set and roster notation. So we can do some calculations down here. Let's actually figure out what F of negative four is. Okay? My function rule tells me to take the input and divide by two and then subtract three. So that's going to be negative two minus three, which is going to be negative 5. That's typically shown in one of these diagrams by drawing an arrow and showing that negative four ends up going to negative 5. If I then do F of 6, that's going to be 6 divided by two minus three. That's three minus three, and that gives us an output of zero. So I show this. And finally, if I do F of ten, clock at ten divided by two minus three gives me 5 minus three, and that gives me two. Now, it's important. The range here can not be stated as either negative 5 less than or equal to Y less than or equal to two. Certainly we can't use interval notation either. So I want to change to red. I'm going to go like this. And put a giant X through it. The reason I can't do that and listen to this carefully is that if I have an answer, let's say like this one or this one. What that implies is that any real number between negative 5 and two lies in the range. But in fact, the range actually only has three numbers in them. So we list them with these curly brackets. This is what's called roster notation. We just list them negative 5, zero, and two, negative 5, comma zero, comma two. All right, we just list them out. There's our range. We can't write it like this. Otherwise, it would include things like Y equals one. Or Y equals negative three, right? Those are between negative 5 and two and yet they're not in our range. All right? So a little mapping diagram, little finite domain, finite range. Okay, I'm going to clear this out. So copy down anything you need to. All right, scrubbing. Excellent. Let's move on to the next problem. Okay. Nice little multiple choice problem here. Which of the following values is not in the domain of the function F of X, shown below. Illustrate your thinking by marking points on the graph. Okay? So I'd like you to pause the video right now and see if you can answer this question. All right? All right, let's go through it. So it's important. When working with domain and range, the most essential thing to remember is that the domain is X and the range is Y now we're talking about the domain here. So I'm going to cross off range. So if negative three is in the domain, we should be able to go to X equals negative three and see a point. And there is one. So that's the wrong answer because I'm looking for a value of X not in the domain. Let's go to choice two, negative four, two, three, four. Here is negative four, and there's a point. It may not be a nice point, but it's a point. Let's go to choice three. 5, one, two, three, four, 5. Oh, there it is. Right? That's not in the domain. Because the graph doesn't extend that far. Side note, let me just do the last one. So there's our answer. But in X equals zero, it would be right here. So that would work fine. All right, the most important thing about exercise three is knowing that domain or the X values. We wouldn't want to be looking at the Y values here. Okay. I'm going to clear this out, and then we'll go onto a last problem involving our friend, the piecewise function again. Okay, I'll clear it up. Let's take a look at a piecewise function. All right, piecewise linear, again, linear just means a line, okay? Looks complicated, but don't let this fraction scare you off. It won't be too big of a deal. The question just simply asks me to determine the function's range. Now, the range, as we saw in the last problem, the range. Is the set of all the Y values Y values. But it's hard to know what the Y values are because I haven't produced a graph of this yet. So what I'm going to do is I'm actually going to use my graphing calculator to create this. Now we just learned how to use the graph and calculator to explore functions recently. So we're going to use it again. So right now I'm going to open up the TI 84 plus. There it is. I love my TI 84 plus. Now, we haven't really used it yet to graph piecewise functions. And what's really neat is I can now enter both of these formulas in. Produce a table and really look at that table to get those values. All right? So let me hit Y equals. All right. Let me clear out any equations that might be sitting there from past lessons or past problems. So clear both Y one and Y two app. If you have anything in Y three, Y four, et cetera, then be a good idea to get rid of those as well. But we're just going to use Y one and Y two. All right. Now that we have everything clear, what we're going to do is go up to Y one. And we're going to carefully enter that formula. So use that negative symbol, don't you subtraction, put in that parentheses, put in X plus two, close the parentheses, and now divided by two, okay? Look at the formula, make sure it looks good. All right. Now let's go down to Y two. Now I do want to emphasize. There's only one function here, but we're going to put them in Y one and Y two to help us out. So now in Y two, maybe a little bit easier, let's do four times X minus ten. Okay, check over our formula. Make sure it's right. All right. Now what we have to do is we have to create a table, right? We want to create a table of values for this thing. And we want to watch which formula we use. The first formula we're going to be using from negative four to two, and the next one from two to four. All right? So let's do this. Let's go into our table setup. Remember how to do that. It's right up here. So let's go into table setup. And this is where we need to start our table. Since our first formula is defined from negative four to two, let's start our table at negative four, and let's go by ones. Okay? Now, let's pop in to our table and see what it looks like. All right. Great. We're in our table now. Okay, and notice that we've got an X column, a Y one column, and a Y two column. Okay, the Y one column was that first formula, the Y two column was the second formula. We want to read off the first column all the way up into X equals two. All right? What I'm going to do is I'm going to put my values down and I'm going to put them right here in a table. I'm going to make my table horizontal. I know that the tables on the calculator are vertical. But I'm going to have a little more space here. If I go horizontal, so I'm going to do X and Y I'm going to start at negative four. And then in the table, what I can see is I can see that my Y value there. That negative four is going to be equal to one. Then when I see is at negative three, my Y value is 0.5. At negative two, my Y value is zero. At negative one. Sorry, I have to do a little bit of a cheating here. It's at negative .5. At zero, it's at negative one. At one, it's at negative 1.5. And at two, it's at negative two. Now that's where the other formula takes over. But take a look at your table. Notice in Y two at X equals two, the Y value is still negative two. So that's a good thing, right? It's about those formulas meeting up, which they don't always do in a piecewise function. But so far, that's what we've been seeing. All right, when I put three in though, and I read off of Y two, so Y one applies kind of from here back and then Y two is going to go from here on. When I put three in there, I find that I have two and when I put in four very far that that thing extends, I have 6. All right. So now this allows us to graph this function. All right, I'm going to go down negative four and go up to one. Negative three and .5, negative two, and zero, negative one, negative .5, zero, negative one, one, negative 1.5, two, negative two, right. Three, now again, that's where that other formula would take over. But it has the same Y value. At three, I'm at positive two. And at four, I'm at positive 6 two three four 5 6 all the way up here. So I'm going to just connect these things, I think, with a couple of blue lines. One there. And one there. Now I want to really emphasize something right now. Okay. No. Arrows. All right. I also don't need my graphing calculator anymore. So let's get rid of that. All right, bye bye TI 84. I don't want to put any arrows on because the function doesn't extend any farther. In fact, the domain of this function, I know that's not what the problem asked for, but the domain, the X values go from negative four, all the way to positive four, but nowhere else. So now we can really look at the range the Y values. Okay? And what we really want to do is find the smallest Y value, which occurs right down here right at the bottom. What is that? That's at negative two. And the largest Y value is way up here. And that's a positive 6. And there's our range. Again, another nice way of doing it with interval notation, negative two to 6, with the brackets, because they both occur there. All right? And that's it. So a little bit of practice using our graphing calculator to generate tables of values a little bit of practice on piecewise functions, all in order to find the range. Y values get hit. That's the range. All right, I'm going to clear this out. So think hard about what we just did, the practice we got on the calculator. All right, here we go. All scrubbed. Let's finish up this lesson. All right, so in unit three lesson number 7, we looked at the domain and the range of a function. The domain being the set of all inputs that give us outputs and the range really being the set of all outputs. A lot of people will think about the domain as being the X values and the range as being the Y values. And that's a great start. There's nothing wrong with that. And we'll keep reinforcing this as we go on. For now, though, I want to thank you for joining me for another common core algebra one lesson by E math instruction. My name is Kirk weiler, and until next time, keep making. I keep solving problems.