>> Suppose f of x is 4x minus 1, g of x is x plus 5. Now, it turns out that we can actually perform an operation, a new operation where we put one function into another and this is called the Composition of Functions. And it works like this, the notation is f of g of x that we're putting the g function into the f function, this is the notation. Now, let's just kind of go through with the bracket notation here, working inside the bracket we would replace g of x with the expression for g of x, so this would be f of x plus 5. Now, this is in the f function replace the x with x plus 5. So, we're replacing x with x plus 5 in the f function, so instead of 4x minus 1 it's 4 times binomial minus 1. So, 4 times x plus 5 minus 1 and then we will clean this up a little bit 4x plus 20 minus 1, 4x plus 19. Alright, so that's the idea. Now, let's go the other way, let's compose the two functions in the other direction that is let's put the f function into the g function. So, it's g of f of x and following the notation it's g of, well what is f of x? f of x is the 4x minus 1. And then g of that means in the g function replace x with that binomial. Here's the g function replace that x with this binomial and we would have 4x minus 1 binomial plus 5, so 4x plus 4. We can even put a function sort of into itself, that is f of f of x means we're going to put the f function into itself. So, it is f of, now, f of x is 4x minus 1, f of that binomial means in the f function replace x with binomial. So, the f function is 4x minus 1, so it's 4 times 4x minus 1 minus 1, we clean it up we get 16x minus 5. Now, there is another notation for the composition of functions and I'd like to show that notation to you. This is the easiest one to follow, but another notation for this is this and I'll use it in all three places. This would be f of g of x would be f of g of x like this. Now, this is not a zero and it's not an o, a letter o. It's just a little bitty circle kind of thing. And the students read this as [inaudible], but f of g of x is what this is, that's the notation for it and this is the equivalent notation and I suggest that if you see this you write that, because it's easier to progress through this than it is that, because we're unfamiliar with this somewhat. Now, this one would be gaf x or g of f of x. And when we're putting f into itself it is f of f of x like this, but we would write this and proceed from there.