Common Core Algebra II.Unit 2.Lesson 3.Function Composition

So let's jump right into it. Exercise number one, a circular garden with a radius of 15 feet is to be covered with topsoil at a cost of a dollar 25 per square foot of garden space. Letter a asks us to determine the area of the garden to the nearest foot. All right, so we're going to determine the area of the garden, and of course we all know how to find the area of a circle. The area of the circle is given by pi R squared. So if the radius is 15 feet, then the area of the circle is going to be pi times 15 squared. Because we want this to the nearest square foot, not in terms of pi, which kind of makes sense because we're doing something with it that is practical. Then we would just multiply this all out on our calculator. It's kind of a messy answer. It turns out to be like 706.858, et cetera, and that ends up being 707. Square feet of area. Now that was a function, right? The input was the radius. And the output was the area. The input was the radius and the output was the area. But let her be now says using your answer from letter a calculate the cost of covering the garden was topsoil. 

Well, we know that the area was 707 square feet. And we're going to multiply that by $1.25 per square feet. Remember how units cancel. And that's going to give us 800 and $83. And 75 cents. So think about this for a second. A radius of 15 went into a function. That function being the area function. And what came out was 707 ft². Then that 707 ft² went into the cost function. And what came out was 883 75. I didn't really care that much about the 707 ft². What I ultimately really wanted to know was how much it was going to cost me to do this. All right? And that's the idea of composition. An input going into a function, giving us an output, et cetera. So let's take a look at what that kind of looks like in a diagram. I'm going to have you pause the video now, write down anything you need to, and then we'll move on. Okay, here it goes. I think we'll switch back to blue. 

So function composition, very simple. We always start with two functions. Or more, we could have three, four, 5, doesn't matter. But at least two. You have some initial input, and it goes into F of X now, what always happens when you have something that goes into a function is something comes out of a function. And the output that comes out of F of X suddenly becomes the input to G of X and then just like in our last problem, we get some kind of final output. Right? So the idea of function composition is very simple. It's like a row of dominoes knocking each other over. One after another. Okay? Let's take a look at how it looks in symbolic form. All right, so we've got two functions. One of them, the rule is F of X equals X squared -5. And the other one, G of X equals two X plus three. We want to find values for each of the following. So how in the world do we even interpret this? Well, it's actually not that bad, because G of one, this thing right? Inside of here, that's got a value. G of one is, well, two times one plus three. So G of one is 5. But equals can always be substituted for equal. So F 5 then is 5 squared -5, which is 25 -5 is 20. And that's it. F of G of one is 20. Notice how I said that. 

Let me do a little racing here because this got a little bit busy. I said F of G of one. So I've got G of one, which sits inside of F it is the input to F now, it's not that big G has to come first in F has to come second. In fact, if you look at this, G of F of two, it implies we really have to figure out F of two first. Which would be two squared -5, which would be four -5, which is negative one. I now need to know what G of negative one is. That's two times negative one plus three, and that's negative two plus three. And that's positive one. That almost looks like a one. That probably looks less like a one. Oh. That's supposed to be a one. G of G of zero. We can have that. We can definitely have an input go into a function. Give us an output. Take that output, put it into a function. Over and over again. So that's all this is saying. This is saying, all right. 

Well, G of zero, it's going to be two times zero plus three. So it would be zero plus three. Which is three. And then that goes right back into function G again. Now, I think the concept here is pretty easy. What can get a little bit trickier is another type of notation. Some people call this fog notation. Not because it makes you foggy, but literally F of G of negative two. So we know we've got the two functions. The question is what where does the negative two go first? And this is pure convention. The negative two goes into this one first. So G of negative two, two times. Sorry about that. Two times negative two plus three. Two times negative two is negative four. Plus three is negative one. So that negative one now has to be fed into F so F of negative one is going to be negative one squared -5. Gives me one -5. Gives me negative four. 

All right. So that's a little bit trickier. The previous notation was pretty easy because, hey, whatever the input is sort of inside first, that's the first function you deal with. Here it's way more questionable, you know? But again, the three will go into this F, so F of three, three squared -5, 9 -5 equals four, then four gets fed into G of four, two times four plus three. Gives me an 8 plus three gives me 11. And again, we can always compose a function with itself. In fact, there's many situations where that makes sense. So the first thing I'm going to do is put that negative one into F that's going to be negative one squared -5. One -5 is negative four. That negative four then gets looped back into F. And we'll get 16 -5 and 11. All right. So composition. Input goes into a function output comes out, that output goes into another function and a final output comes out. It's not too bad. Pause the video now and write down anything you need to. Okay. Clearing out the text. 

Now, if we can do it with formulas, we can do it with graphs. Keep in mind that the inputs are X values and the outputs are Y values. So here in this graph, we've got Y equals F of X, and in this graph, Y equals G of X so those are going to be our rules, given graphical form. Let's take a look at G of F of two. Well, the first thing I had to do is figure out what F of two is. In order to do that, I go to the graph of F, I go over to X equals two, and I find the Y value is three. That means I now need to figure out G of three. So I go over to this one, two, three, and I find out that G of three is three. All right, I'm going to erase some of that stuff because I wanted to throw you off. Let's try F of G of negative one and then have you work on the rest of them on your own. So first thing I have to do is figure out G of negative one. Come in here, here's negative one, one, two, three, four, 5, G of negative one is negative 5. I now need to figure out F of negative 5, so one, two, three, four, 5, and it looks like that's going to be negative. One half. All right. Why don't you work on the rest of them on your own? All right, let's go through them. G of G of one. 

All right, well, the first thing I want to know is what's that G of one equal to? And it looks like it's equal to three. Now I want to know what G of three is. So one, two, three. And it looks like it's three. All right, now we go to this notation. Remember the negative two goes into F first. So F of negative two is positive one. G of one, then, is positive three. Love those threes in this case one, two, three of them so far. All right, here I want to go zero into G, G of zero is its Y intercept. Zero zero. Look at that. Corresponds to this point right here. F of zero Z Y intercept. And that's two. Huh. Wow. Looks like we just dealt with F of zero. So now we get to deal with it again. And so we just saw F of zero is two. And F of two is our favorite three. Lots of threes here. All right, so we should be able to do function composition if we're given graphs. Hey, we should be able to do it if we're given a graph and an equation or a table in a graph. We just need to have two functions such that when I give you the input, you can figure out the output. All right, I'm going to clear all this out. Okay, now, sometimes what we want to do is come up with a formula for composition. So that we don't have to do it for each individual X now, this can be a little confusing. 

So let's take a look at it with a linear, a pair of linear functions. F of X equals three X minus two, G of X equals 5 X plus four. It says determine formulas and simplest Y equals AX plus B form for F of G of X while strangely enough, we really don't do anything different than what we were doing. If I put X into G, I get 5 X plus four. Now I'm going to write down something that you haven't seen very much of. I now want to know what F of 5 X plus four is. Now, what is F of X? Well, the rule says take the input, multiply it by three and subtract two. Take the input, multiply by three, subtract two. But now the input is an X it's 5 X plus four. So I'm going to take that input. I'm going to multiply it by three. And I'm going to subtract two. Doing the distributive property that would give me 15 X plus 12 minus two. And 15 X plus ten. All right, see if you can get a formula in letter B for G of F of X. All right, let's go through it. Well, again, if I put X into F, what I end up having coming out is three X minus two. And then if I want to know what G of three X minus two is, keep take a look at G, G's rule is take the input and multiply it by 5. Here are the input those three X minus two. And then add four to it. So let's familiar 15 X minus ten. That's not so familiar. Plus four. And that's going to be 15 X -6. Notice these two formulas are not. The same. And they rarely will be. Therefore, composition is not commutative. It matters which function comes first and which function comes second. 

All right, we'll pause the video now and write down anything you need to. Okay, let's clear it out. Maybe do one more problem. Nice multiple choice. If F of X equals X squared and G of X is X -5, then what is F of G of X? Think about it a little bit. Well, this one's tricky on a variety of different levels. But you have all the math you need. Keep in mind that G of X is X -5. So F of G of X will be F of X -5. Now what's F of X? It's take the input, which is X -5. And square it. Now you gotta watch out on here. Right, multiplying two binomials can't be something that you know how to do on a Monday, and then forget by a Tuesday. Very important. That should be a 5. Looks like a 6. So all I'm doing is multiplying out. What they told me, I would have to multiply out. I said X -5 squared. So I get X squared minus ten X plus 25. All right, a little tricky there. Easy, easy to go with choice one or choice two. All right. Well, I'm going to clear out the text. It's gone. 

Let's finish up. So I think that the idea of function composition is pretty easy. An input goes in, output comes out, that output goes into another function, final output comes out. Pretty easy, but of course we can deal with it in terms of tables, graphs, equations. And then there's that tricky piece at the end, which we just started taking a look at, which was how do you do function composition and come out with a formula instead of just numerical values. More on that later. For now, I'd like to thank you for joining me for another common core algebra two lesson by email instruction. My name is Kirk weiler, and until next time, keep thinking. And keep solving problems.