Common Core Algebra I.Unit 3.Lesson 1.Introduction to Functions.by eMathInstruction

Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk weiler, and today we'll be doing unit number three lesson number one, introduction to functions. Before we begin this very, very important topic and lesson. Let me remind you that you can find the worksheet that goes with this video, along with a homework set by clicking on the video's description or by visiting our web page at WWW dot E math instruction dot com. Also, just to remind you about those QR codes at the top right hand portion of our worksheets, use your smartphone, or a tablet, scan the code and come right to this and other videos. Great. Let's begin. Hey, there's my QR code. All right, now let's begin. The definition of a function. Almost all higher level mathematics is based on the concept of a function. So understanding its very elegant and very simple definition is important. Let's take a look. A function is a clearly defined rule that converts an input into at most one output. All right? Often we think about this as just being kind of like a box. All right? We got some box represents our function and input goes in, very often it's an X value, and a single, and this is important. A single Y value comes out. All right? At most, no more than one output. The reason that we can't just say, hey, it converts an input into one output is that sometimes if you put an input in, you don't get an output out. For instance, if I asked you how many days were in the month of Illinois, you'd have to say that I was crazy. Because that input doesn't even make sense, and therefore you can't give me an output. All right. But when there is an output for an input, there can be only one. Oh. Now, excellent. These function rules have to be clear, all right? They must be clearly defined. And the way that they're defined often with numbers is in four different ways. Some people even call this the rule of four. Very often they're defined using equations. Sometimes they're defined using graphs. Very often they're defined using a table of some type. And sometimes they're just defined using some kind of a verbal description. All right? So we're going to get use with all of these different forms, and in some problems, we're going to be using more than one of them. In fact, in the first problem, we're going to really put ourselves through the motion and understand functions. These rules, in all these different ways. Okay, I'm going to just clear out those little blue lines. And let's jump into the first exercise. Okay. Exercise number one, consider the function rule. Multiply the input by two and then subtract one to get the output. All right, so let's try to understand this rule. Right now. Multiply the input by two, no problem. And then subtract one to get the output. I think I can do that. Let's generate a table that shows a little bit of this rule being applied. Letter a says fill in the table below for inputs and outputs, inputs are often designated by X and outputs by Y so what does it say? It says take the input and multiply it by two. And then subtract one to get the output. So two times zero, zero. Minus one gives me negative one. All right, so an input of zero gives me an output of negative one. All right, let's go with an input of one. Right? If we have an input of one and we multiply it by two, and then we subtract one, that's pretty easy. Two times one is two. Minus one gives me an output of one. Sorry. It doesn't quite look like a one. That looks a little bit better. When my input is two, and I multiply it by two, and then subtract one, I get four minus one. Which gives me an output of three. And then when I multiply my input of three by two and subtract one, I get 6 minus one. And I get 5. Right. So that's simple enough, right? Inputs outputs. That's it, right? And only one, notice we only had one output for each of those given inputs. In, out. Now let her be says, write an equation that gives this rule in symbolic form. Okay, well here we go. Here's my output. Now, how do we calculate my output? Well, in each time what happened was we took our input, here I'm going to underline our inputs with this red. We took our input each time and we multiplied it by two, we took our input, we multiplied by two, and then we subtracted one. So there is the symbolic rule for our function. There it is. And finally, we can also graph it. It says use your table in a to help. Okay? Now we haven't done a lot of graphic, but it's pretty easy to create coordinate pairs here, right? Remember, exomes comes first, so we have the coordinate pair zero comma negative one. Coordinate pair one comma one, coordinate pair two comma three, and the coordinate pair three comma 5. Again, this is where I'm getting the X coordinates. And this is where I'm getting the Y coordinates. So when we plot the zero negative one is here, one one is here. Two, three, right here. And three, one, two, three, four, 5. This right here. We could keep going, right? If we had four, if we wanted the output for an input of four, just to see if we could fill it up, that would be 8 minus one or 7. So we'd get four comma 7, which would be right here. I can actually. Draw that. Uh oh. That's not the way I wanted it. I know there's no real way of getting rid of that. Besides doing that. Here we go. I'll just manually draw the arrow on it. All right. And there's our rule, right? In graphical form. Of course, I'm connecting these dots because we can have inputs that are non integers, that aren't whole numbers. We could have an input of one half, which would actually give me an output of zero. Now notice on a graph, right? And just in general, inputs are thought of as the X coordinates, outputs are thought of as the Y coordinate, and that means for every X coordinate, there can be only one point at most one point. You can only have one Y for a given X all right, and this was a very, very full exercise. So pause the video now, think hard about what we did, all right? Write down what you need to. All right, here we go. All cleared. Let's keep moving on. Exercise two. In the function rule, from number one, what input would be needed to produce an output of 17. All right, now, this is kind of cool, right? It says, it's actually asking for the input that I would need to get an output of 17. Now, I'm going to write down our rule in symbolic form. It was two X, Y equals two X minus one. The verbal description was multiply the input by two, and subtract one to get the output. I'd like you to pause the video right now and do some problem solving. Try to figure out what input you would need to get an output of 17. All right. Well, the problem didn't specify how you had to do this. So if you played around with it with guess and check, I think that's awesome. If you tried to extend the graph that we made in the first exercise using a larger piece of graph paper to try to find a Y coordinate of 17, that's great as well. I'm going to go with sort of the easiest approach that I know of, which is to take my output and make it into 17. See, then I can use algebra to find that input, right? So I know the output is 17. So I'll solve this equation by using properties of equality. The additive property of equality allows me to add one to both sides. And then the multiplicative property of equality allows me to divide both sides by two. So we need an input of 9 to get an output of 17. Now, I think that the second question is almost as important if not more important than the first. It says, why is it harder to find an input when you have an output, then finding an output when you have an input? And I know there's a lot of input output output input stuff in this question. But think about it. Think about how easy it was when I gave you an input of X equals two to find an output of Y equals, I guess it was three. And then when I gave you an output of 17, it's actually kind of hard to find an input of 9. Why is that? Well, I'll write this down in a second. But it basically boils down to the following. All right, function rules tell us how to convert an input into an output. They do not tell us how to go in reverse. They do not tell us how to find an input if we're given an output. So. Function rules and they do rule, function rules. Tell us. How to convert. And input. Into an output. Not the other way around. Now, in algebra two, you're going to learn about things known as inverse functions. And inverse functions do that. They tell us how to undo what an original function has done. But for right now, we're not going to be working with inverse functions. We're just going to be going in the forward direction to go in the backwards direction to find the inputs if we're given outputs will always be harder. But that's okay. We'll find ways to do it. All right, I'm going to clear out the text, so copy down what you need to. All right, here it goes. Moving on. Exercise three. A function rule takes an input N so it's just what we're calling it. This time it's not X and converts it into an output Y by increasing one half of the input by ten. Determine the output for this rule, when the input is 50, and then write an equation for the rule. All right, pause the video now and see if you can do both of these things. Start with the verbal description of the function. Try to use the verbal description to change that input of 50 into an output. And then based on what you did to the 50, write a rule involving the Y and the end. All right, let's go through it. So let's take a look. Verbal description of the function, right? That's what we're given first. What do we do? We're going to increasing one half of the input by ten. All right, well, let's start with the 50. We'll go with the 51st. So we want to find one half of 50. That's easy enough. That's 25. And then we want to take the 25, and what do we want to do? We want to increase it by ten. So, in other words, if we think about our little function diagram, when 50 goes into our function, we're going to be great to have some notation for functions. What comes out is 35. Now the question is, can we do the same thing with our box, our fun, fun, function box if we put N in what comes out? Well, we've got to write a function rule here, and that's going to be an equation, an equation, an equation, and an equation has to have an equal sign folks don't forget it, okay? Now, what gives us our output? Well, we take our input and we find one half of it. We could either multiply by one half or divide by two, your choice. And then we're going to increase it by ten. That doesn't even look like an N let me let me make a little bit of a better pen for you. So there it is. There's our function rule in equation form. We've been given the function rule in verbal form. We translate it into an equation form. The beautiful thing about the equation form, obviously, is that it will allow us to convert very quickly various inputs into outputs. If we needed to. All right? So I'm going to clear out this text, pause the video now if you need to. All right, here it goes. So let's keep going. We want to work with functions in as many different ways as we can. All right? And remember, at its base essence, all the function is is a rule that takes an input and gives us an output. Those inputs and outputs are mostly numerical, mostly numbers, but they don't have to be. So let's take a look at one in exercise four. That is not numerical. Function rules don't always have to be numerical in nature. They simply have to return a single output for a given input. The table below gives a rule that takes an input as an input, a neighborhood child, and gives as an output the month he or she was born in. Letter a why can we consider this rule a function? Notice the little type out here will get rid of that eventually, and the worksheets. But why can we consider this rule of function? Think about this for a moment, pause the video if you need to. All right, let's talk about it. Well, for each child, and let me just put this in parentheses input. There is only one birth month. And that would be. Alpha. Right? I was born in December. I wasn't born in December and January. So if the input is the child, the output is the birth month. There is only one output for each input. Letter B is easy. What is the output when the input is rosy? Simple enough, right? The output is February. All right. So when Rosie goes into this function, what comes out is February. Finally, letter C says find all inputs. So here we're going in reverse. Find all inputs that give an output of May. All right? Pause the video right now. And do that. And also, try to answer why that does not violate the definition of function even though there's two answers. Pause the video now and see if you can handle that. All right, let's go through it. Well, what are we looking for? We're finding all the inputs that have an output of May. While there is an output of May, there is an output of May, and now we're going to go backwards, and we're going to figure out that Zeke. And Nico are valid inputs that give us an output of May. Now, the it seems like that shouldn't be a function, and yet letter a, we said, no, no, no, this is a perfectly good function. The reason that we can still consider this a function is because a function does not insist that each input has a unique output. Okay? Outputs can be repeated. And that's the important thing. Outputs can be repeated. In functions. Inputs can not. Inputs can not. All right, if there's no problem with an output being repeated. All right, so let me clear out the text. Pause the video now if you need to. All right, let's go. Let's do it. Okay. This last problem, we've got a beautiful function given by a graph. Let's take a look at what the physical scenario is. Charlene heads out to school by foot on a fine spring day. Her distance from school in blocks is given as a function of time in minutes she has been walking. This function is represented by the graph given below. Let's answer some questions. And actually, what I'd like you to do now, because you've done a lot of work in the past with just looking at and interpreting graphs. Is I'd like you to pause the video. Spend up to maybe even ten minutes taking a look at this graph and the questions that are asked about it. And see if you can come up with all the answers. All right, and then we'll go through them one by one. All right, let's go through the problem. How far does Charlene start off from school? All right, well that really gets at the idea of where was she at at zero minutes? Well, if I look up and watch out, the scale is by twos, right? That could be a little tricky if you don't look at it carefully. It's two, four, 6, 8, ten. Well, it's clear that she starts off 22 blocks, don't forget your units. Starts off 22 blocks from school. Simple enough. Notice only one output. Letter B, what is her distance from school after she's been walking for four minutes? Well, here's our four minute input. And it appears ten, 12, 14, that she is now four T blocks. From school. She's making progress. It's a long way to walk to school, right? Depending on how you look at it, 22 blocks is, you know, it's over a mile. But hey, see that or the bus, right? Letter C, after walking for 6 minutes, Charlene stops to look for her subway pass. How long does she stop for? All right. Well, here's our 6 minute time. Now, how can I tell that she's stopped? I can tell that she's stopped because at 6 minutes she's ten blocks away from school. At 7 minutes, she's ten blocks away from school. At 8 minutes, she's ten blocks away, et cetera. In fact, she remains ten blocks away from school. Between 6 and 9 minutes. So for three minutes, right? For three minutes, she stopped. And that symbolized by sort of this, this and this. Letter D Charlene then walks to a subway station before heading to school on the subway. A local. How many blocks did she walk to the subway? Well, let's take a look. Now as she's riding the subway to school, she's going to be getting closer to school. But if you've ever been walking down a sidewalk, let's say, I don't know, in Brooklyn or something like that. And you pause, you might realize that you overshot the subway stop, right? So maybe you have to walk a little bit away from the school simply to get to the subway stop. So it's at this point, right? That she's actually traveling on the subway. This is when she's on the subway. So how many blocks did she walk to the subway? Well, right here, she said ten blocks right here, she said 12 blocks, so she must have walked two blocks to the subway. Now, how long did it take her to get to school once she got on the train? Well, she must have gotten on the train at 11 minutes. And then she was at school at 15 minutes. The time that elapses between 11 minutes and 15 minutes. Is four minutes. I know this is idealized. I know students and teachers who are looking at this and they've been riding the a or the three or the two or the 9 and in New York City realized that, okay, you're probably standing on that subway platform for a while before you even get on the train. But still, that would make for one complicated graph. All right? Anyhow, I'm going to be clearing out the text, so pause the video now if you need to. Here it goes. All right, let's wrap up this lesson, shall we? So today we introduced quite possibly one of the 5 most important ideas or concepts in all of math. That of a function. A function is a rule doesn't have to be simple. It doesn't have to be complicated. But it has to be clearly defined. It clearly defined rule that takes inputs and converts them into at most one output. One output. All right, we're going to obviously be working with functions a lot, lot more. And in fact, this entire unit is devoted to them. So thank you for joining me for another common core algebra one lesson. My name is Kirk weiler. And until next time, keep thinking. And keep solving problems.