 In this last video for lecture four, we're gonna change pace just a little bit. We've been talking a lot about function composition. And in this video, we're gonna talk about something called function decomposition, right? So we wanna break apart the composition that we have been doing. So what this means is, we are gonna be given a function. In this case, we have capital F of X equals the fourth root of X plus nine. And we wanna find two functions F and G so that when F is composed with G, we get back capital F. That is, can we recognize big F as a composite of two simpler functions? And it turns out there's a couple of different ways one could do this. This is not a unique process. So what approach would be the following? We take F of X to be the fourth root of X, and we take G of X as X plus nine. Now notice, if we take F composed with G of X, we're gonna take the fourth root of X and put G of X inside of it. With G of X, of course, is X plus nine. This gives us back capital F. So this, in fact, then works out perfectly here. But compared to an earlier example, one of us might be tempted to do something like the following. We could take F of X to equal the square root of X, and then G of X could be the square root of X plus nine. In which case, we then get F composed with G if we put this back together, we're gonna get the square root of the square root of X plus nine, which is then equal to the fourth root of X plus nine. That works. And so it turns out this composition is not exactly a unique process. It can be done in multiple ways. Now there's always sort of like a cheater way of doing it. If I were to take F of X to equal the identity function, and then G of X equals F of X here. Notice in this situation, if you take F composed with G of X, this is just gonna give you, well, F of G of X, right? But as F is just the identity function, this will just give you back G of X, which itself is just F of X there. So that way works, but again, it's kind of like a cheater method. It doesn't really, because the identity function really isn't offering much. The point of decomposition is for the functions to kind of share the weight, right? So no one function's holding all the weight, which is happening exactly here. You can also go the other direction. You could do little F of X equals capital F of X, and G of X is the identity. That's also a decomposition, but it's a non-trivial decomposition. It's like factoring 12 as one times 12. Although that's true, that doesn't really help us here. What we prefer would be something like three times four equals 12, or two times six equals 12. We actually got numbers that are smaller than 12 inside of the factorization. And so that's what we're doing right here. We're basically trying to factor the function, but we're using not multiplication, we're using decomposition as our factorization idea here. Let's look at another example of such a thing here. Let's take the function capital H this time to be one over X plus one. Can we decompose this as two functions F and G? This one might be a little bit more funky, but again, there's more than one way of doing this. We could take F of X to be one over X, and we could take G of X to be X plus one. And notice when you put these functions back together, F of G of X, this will be one over G of X, where G of X is X plus one. And so this is exactly capital H of X here. So we can do this in multiple ways. And so what you're gonna be asked to do, what I'm asking you to do here is to practice this on your own that when you're given a more complicated function, can we decompose it into smaller functions? There's a lot of benefit of doing this. What we're gonna see next in our lecture series is that decomposition of functions is actually what graph transformations is all about. Shifting a graph left, right, up, down, this all comes to composition of functions. And therefore, if we can decompose functions, we can identify the transformations of the graph in play here. But also even more important than that, this idea of function decomposition is the key ingredient for the chain rule in calculus, which is a topic many of you will study in the future. Being able to use the chain rule comes down to being able to decompose functions. And as I said many times already, college algebra or pre-calculus courses are often the karate kid of mathematics here because we're preparing you. I mean, yes, these skills are available now to help you with many real life applications, but many of these skills might feel out of context. And that's because the context is calculus. We're learning to wax the car and paint the fence, not to make Mr. Miyagi's house look better, but to practice karate, which at the moment we don't exactly see it because we don't have the Cobra Kai kicking our butt at the moment. So this idea of function decomposition is a skill that's very useful to practice now in algebra, but also we will be beneficial for us when we move on to a calculus type problem in the future.