Common Core Algebra I.Unit 4.Lesson 2.Function Notation.by eMathInstruction
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Learning Common Core Algebra I Unit 4 Lesson 2 Function Notation by eMathInstruction
Welcome to another common core algebra one lesson by E math instruction. I'm Kirk Weiler, and today we're going to be looking at unit number three lesson number two on function notation. As always, you can get a copy of the worksheet that we go through in this lesson and in the company homework by clicking on the video's description. All right, let's talk a little bit about functions before we launch into function notation, whatever that is. Remember from the last lesson, a function is an extremely useful device in mathematics. It's something that essentially takes an input, very often X and gives us an output, oftentimes Y and the key is, for any input that can actually go into the function, there is at most one output. At most one, not two.
I was kind of picture functions as being kind of analogous or similar to those big gumdrop gun ball machines, right? The ones that give you the big piece of gum, you know, you take your quarter, that's the input, you put it in, you crank. It's almost like the crank and the machine is the function and what pops out. That big piece of bubble gum, right? That's the output. Okay. Now, today, what we're going to do is we're going to learn something about functions called function notation. And this is just simply a new way of saying, here's the rule. The rule is what you do to the input to get to the output. Here's the rule. Here's the input, what's the output? So that's what the function notation is going to get at. Let's take a look at the first exercise. Now, I'm actually going to do some function notation problems in this first exercise just to get you used to what they look like.
All right, So take a look at letter a now the way that I read that equation. And let me specifically show you what equation I'm talking about. The way that I read this equation is I call this F of X and what that means is that F is a function and it's a function of the input variable X that's the idea. F of X and then right here it tells you what the rule is. In other words, the rule is take the input, which is X multiply it by three, get a result right, order of operations, and then add 7. Okay? Now, when I see something like this, I would think of this as F of two. So what is the function when the input is two? What is the output when the input is two? So wherever there's an X in this function, I'm now going to be replacing it with this number. So, and more appropriately, it's not so much that I'm replacing it with that number. I'm doing to the input what the rule tells me to do. Take the input, which is two, multiply it by three, I'll get 6, add 7, and I get 13. So that is my output. So what I know now is that when my input is two, my output is 13. F of negative three, wherever that X is and whatever is being done to it, I'm going to replace it with negative three. So I get three times negative three, which is negative 9. Negative 9 plus 7 is negative two. So for an input of negative three, I have an output of negative two. It's pretty much that simple.
Now, of course, the reason that function notation is very confusing for students and why don't we illustrate this with letter B is that it appears like we're taking some variable G and multiplying it by some variable X G isn't and variable per se, F, G, H, right? They're the rule. And it's really saying, hey, look, the rule for the input. And I'm going to give the rule the name G that's just what I'm going to call it. Okay? The rule for the input in letter B is to take the input, subtract 6 from it, and then divide that result by two. Right? And so G of 20 means to take an input of 20, subtract 6 from it, and divide the result by two. So we get 14 divided by two and 7. So for an input of 20, we have an output of 7. It's not too bad. Right? G of zero, when I take my input, watch yourself, right? Not 6 minus zero, zero -6, right? That's going to be a negative 6. Divided by two gives me negative three. So when the input was zero, the output. Was negative three, you know? It's like X is your quarter, right? G of X is the gumball sort of like machinery. And the Y, or the output, well, that's your gumball. Okay, I know we haven't really done much with square roots this year, but remember, all the square root is asking you to find is that number that when multiplied by itself, which is known as squaring, gives you the number that's underneath the square root. We call that the square roots argument.
So, for instance, the square root of a hundred is ten, because ten times ten is a hundred. So in this particular rule, what it says is to take the input, multiply it by two, add one, and then find its square root. All right, let's take a look. Well, if I put four in there, I'll have two times four plus one oh my goodness. My mind just skipped ahead. Let's go back. All right. I then have 8 oh oh. I need more space for this problem. I'm going to suddenly do it down here. I don't know what I was thinking on my spacing. H of four, then would be the square root of two times four, which is 8 plus one, which is the square root of 9, which is three. That's a pretty complicated function that we had there. Erase some of this stuff. All right, let's do H of zero as well. I'm going to think I'm going to pop that one down here. Just give us give ourselves a little more space. Right again, what does the rule say? It says take the square root of two times my input plus one. Well, anything times zero is zero. Right? So I get the square root of one, which is one. Okay? So these rules, which of course can come in all sorts of different forms.
Now have a way of sort of being expressed so that you can very easily say, hey, here's my input. What's my output? All right. If you need to, pause the video for a second and really look at the text that's on there because I'm going to scrub it, okay? You ready? All right. It's gone. Now let's talk function notation. Why equals F of X? In function notation, there's always sort of three things, if you will, right? There's the page cleared. Let's try that again. There's the input, which we typically think of as X, there's the rule itself, very often given as an equation, but not always, we're going to see pretty soon, it doesn't have to be. And there's the output. Now, one thing I do want to mention real early on because this is kind of key. Notice this equality sign here. I take equality very seriously as my students know, right? That means that the two things are the same. In other words, the output and the rule, if you will, we kind of consider the same thing in mathematics. All right? Because the output is equal to the rule, or it's equal to the manipulation of the input. I don't know.
That's a little bit heavy. Anyway, here's my input. Here's my rule. Here's my output. All right, I'm scrubbing. Let's move on to the next sheet and do another problem. All right, let's take a look at our new rule. Given the function F of X equals X divided by three plus 7, do the following. I think at this point, you are probably good to go in terms of a and B so I'd like you to pause the video and take a moment and see what you can write down for those two, okay? All right, let's go through them. Letter a says, explain what the function rule does to convert the input into an output. Now what I hope is that you, you told exactly what happened and the order in which it happened. And I'm just going to number it. So what I'm going to do is I'm going to take the input, right? And I'm going to divide. By three, right? And then I'm going to add. 7 to the result. Of one. Right, so I step one, I take my input, I divide by three, step two, I add 7 to the result of step one. Right, that way we really kind of walk through our order of operations. And that sets us up very nicely for B so remember, the way that we're going to interpret F of 6 has nothing to do with multiplication.
It just says when the input is 6, what's the output? Well, I'm going to take my input. I'm going to divide it by three, and then I'm going to take the result and add 7. 6 divided by three is obviously two. Two plus 7 gives us an output of 9. That's not so bad. Let's get a little bit of a workout with negatives. Try to do as much of this as you can without the calculator, right? Take my input, divide by three, add 7. Remember when we take a negative and we divide by a positive, we always get a negative 9 divided by three is three. So negative 9 divided by three is negative three. Plus 7, again, get a little bit of a workout, three negatives cancel three positives and we're left with an output of four. All right, so when the input was 6, the output was 9, when the input was negative 9, the output was four. All right. Let's take a look at letter C I'd like you to pause the video and read over that right now. And think what they're asking you to do. All right? Pause the video. All right, take a look. Now, notice what it says. It says find the input, find the input for which F of X equals 13. Now what a lot of students will do in this situation is they'll say, all right, I understand what's going on. And they'll put 13 in for X, but in fact, X is what we're being asked to find. That is the input. They're asking us to find this value. So what's the 13? Well, the 13 is the output. So in fact, we have to solve an equation. We have to ask ourselves where does X plus three X divided by three plus 7? Where does that equal 13? That's all.
Now if you didn't realize that's what you were supposed to do, what I'd like you to do is write that down now. Pause the video and solve the equation. All right, let's go through the solution. Remember, in this situation, all you want to do is you want to look, you want to say, hey, X shows up only once, so I'm just going to reverse the order of operations, X is divided by three, then we're adding 7, divided by three, then we're adding 7, so we're going to undo that by subtracting 7 from both sides. We'll get X divided by three is equal to 6. I'm going to go through this quite quickly because we've covered this in the last unit. I'm going to multiply both sides by three, and I'm going to get X equals 18. So for an input of 18, we have an output of 13. Be careful about that, right? Make sure you know whether you're given the input or the output and which one they're asking for. Now letter D is particularly challenging, but I think that many of you are going to be up to this challenge. I want you to pause. I want you to pause the video and see if you can figure out what the answer to letter D is. All right, let's go through it. Wow. Wait a second. They just gave me a whole nother function. The function G of X and what does the rule all functions are rules? What does the rule for G of X say I should do? Well, it says, I should figure out the output to the function F that's what I should do.
Figure out the output for F, whatever the X value is, I should take that output multiply it by two and subtract one. That's what I should do. So if I want G of 6, what I need to do is find two times F of 6 minus one. Now, if we needed to, we would then use the rule for F to calculate F of 6. Thankfully, in problem B if we go back up to it, we see that we know what F of 6 is, F of 6 is equal to 9. So we can just say that's two times 9 minus one, 18 minus one, 17. So there's a nice challenge problem for those of you that learn very, very fast and you're already comfortable with function notation. All right, pause the video for a second if you need to, and then I'm going to scrub the text, okay? And it's gone. Let's move on to the next problem. Now, functions come in three primary forms, equations, which is what we've been dealing with so far in this lesson. Tables, which is what we're going to deal with next, and finally graphs. I absolutely love tables when it comes to function notation. Because they are seriously easy to use. All right, exercise number three says that we've got boiling water at 212°F. I'm going to see if I can move this out of the way. There we go.
Now we can see it. I have boiling water at 212°F, and it's left in a room that is 65°F, and begins to cool. So you can kind of imagine what's going on here, right? We've got, I don't know, we've got some something going on, you know, if we were to graph this, let's say the temperature versus the hours, right? Probably be getting cooler and cooler. Something like that. But we don't have that graph. And we don't have an equation, what we have is a table that gives us the temperature. Here we are. As a function, look at what the input is. A function of the number of hours that the fluid has been cooling. All right, the boiling water. So the first thing it says is figure out what T of two is. Now that's kind of interesting. In the last few problems, when we were trying to figure out functions, you know, we were given an input. And we had to do something with it. We had to divide it by three and then add 7 or take the square root of it after we multiplied it by two and added one. Here it's actually much easier. All I'm asking for is when the input is two, what's the output? Well, there it is. T of two is a 104. And by the way, it's got units, right? A 104°F. So what's T of 6? Yeah, you got it, right? The temperature at 6 hours, right? That's what we're interpreting this. The temperature at 6 hours is 68°F. All right, now look at letter B carefully. Pause the video if you need to and see if you can answer that question. All right, did you notice? This time they're giving us the output, right? T of H, the temperature as a function of the hours is equal to 76. How many hours have gone by? So we kind of scan our outputs, here it is. Right? Here's our output of 76.
So now we go backwards and we find that H must be four. And by the way, also has units. Four hours. Units are very important. Never forget that. All right, let's take a look at one more question that's kind of similar to be, but it's a little bit trickier. Letter C asks between what two consecutive hours will T of HB equal to 100. We've heard the term consecutive, right? Because we worked with it before when we worked with consecutive integer problems. Consecutive hours or things like 8 and 9, or one and two. So between what two times will a temperature will T of H be equal to a hundred? Pause the video and see if you can figure this out. All right, let's go through it. Did you say between two and three hours? If you did, you're right. Because if we go up to the table, we notice at two hours. Let me actually write it right down here. At two hours, oh, that's not the variable I used for time, I apologize for that. Let me get rid of that. We used H so when we were at two hours, our temperature was one O four. And then obviously when we were at three hours, our temperature was equal to 85. So somewhere between here, we must have hit 100, right? And therefore the answer is between two and three. Hours. All right.
I like function notation problems when we're given a table, right? Quite easy because the inputs are easy to see, the outputs are easy. Generally speaking, by the way, if you're given a table, and it's got rows like this one, the top row tends to be the input, the bottom row tends to be the outputs. If it's a vertical column, then what happens is it tends to be that the ones on the left are the inputs and the ones on the right or the outputs. Okay, I'm going to scrub out the text, hit pause in the video one more time if you need to, before we go on to the next problem. All right, and it is gone. All right, our last problem today is going to be a function notation problem that involves graph. Why don't you pause the video for a moment and read over just the introduction. I really want you to look at the wording on your own. All right, let's go through it. Exercise three says the function Y equals F of X, you'll often see it said like that. Y equals F of X is defined by the graph below. So this is the graph of F of X it is known as a piecewise linear function because it's made up of straight line segments. We're going to be working a bit with piecewise linear functions this year. That's a mouthful. So I thought I'd get the thought I'd get the terminology in an early on. Now letter a asks us what each one of the other problems began with. What is F of blank? What's F of one, F of 5, F of negative three, F of zero. All right, super easy with a graph.
This is our input, right? But our input is always the X value. And the output output is the Y value. So what I'm going to do is I'm going to go over to X equals one on my graph. And I'm going to go up and I'm going to see one, two, three, four, that the Y value is four. That's all I have to do. The outputs to a function are the Y coordinates on a graph, and the inputs are the X coordinates. Let's do F of 5 together, and then we'll have you do the other two on your own. So again, this is the X value. X is equal to 5. One, two, three, four, 5, go down to my graph. Go across. And I'm at a Y value of negative two. Pause the video now quickly and do F of negative three and F of zero. All right, let's go through them. F of negative three, one, two, three, one, two, three, four. Not bad right, we can just do it that way. Oh, F of zero. Where is X equals zero? It's right here, right? One, two. We actually put that point on there. And that point on there. Because really, what each one of these sort of problems represents is it represents a point on the function, right? That's the .10.
Negative three negative four. In the next lesson, we're going to get into graphing functions. And this is really the key, right? All the pairs here are input output pairs. Input, output. All right, but take a look at letter B as sub a bit different, right? Because here, they're giving you the output. Oh. This is trickier, and notice what it asks us. It says solve each of the following for all values of the input. I'm going to go into red just so that we can cut a contrast with the blue actually, maybe I'll just grab this stuff out. I think I'll still go in red. All right, the output is zero. The Y coordinate. Y coordinate. Wow. Okay, so looking along here, here's a place where the Y coordinate is zero. Here's a place, and here's a place. So what is that? That's an X value of negative one. And X value of one, two, three, four. And an X value of one, two, three, four, 5, 6, 7. That's it. A little bit harder though, isn't it? A little bit harder. It's very easy when you're given an input to get an output because they'll only be one output. But if you're given an output, you're not guaranteed that there's only one input. You try the next problem, where they give us an output of two. All right, did you find the inputs? Let's see, one, two. Here's a Y coordinate of two. And here's a Y coordinate of two.
So what are our X values? Looks like X equals zero. And X equals three. One, two, three. All right. The last question of the lesson, real simple. It asks us, what is the largest output achieved by the function? What is the largest output achieved by the function? Really, really critical. To make the distinction between outputs and inputs, here we're talking about outputs. Now, if we're talking about outputs, might as well stay in red, if we're talking about outputs, then we're talking about Y values. All right. So what I want is the biggest Y value. Pause the video for a second. See if you can figure that out. What's the biggest Y value on this graph? Did you get it? We kind of already saw it once. That tallest peak on this graph is right up here. And what is that? That's it. One, two, three, four. So I can say that Y maximum equals four. This also asks us at what X value is it hit. Well, that's it. At X equals. One. All right, little mixture of colors on the screen there. I like it. So I'm going to pause the video one more time, then we're going to scrub the text. All right.
Let's get rid of the text. Okay. Well, that was our last exercise. I want to thank you for joining me for another E math instruction common core algebra one lesson. This one was unit three lesson number two on function notation. We'll be bringing you more lessons in the future as they become available. For now, remember, you can access the worksheet and the homework for this lesson. By simply clicking on the video's description. Thank you for joining me and until next time keep thinking and keep solving problems.