 back to another screencast about functions where we are instantiating the definitions of all the terminology that are involved with functions here. This example is a little different, although it's also kind of familiar at the same time. I'm using calculus to show us that functions can take on all sorts of different sets and do all sorts of different things, not just things we saw in algebra. So we're going to find a couple of sets here and do some calculus with it. Let's let F sub D, F for the subscripted D be the set of all differentiable functions from the real numbers to the real numbers. These are the sorts of things you deal with in calculus one. All functions whose derivatives are defined at all points. For example, an element of this set would be the function y equals cosine x. That would be an element of that set. And so with the function y equals x squared minus 2, that would be another element of that set. So all the functions that you worked with in calculus when you took derivatives and everything that you might have worked with in calculus, taking derivatives, bundled those all together in a set, and that's what F D is. F is the set of all functions period from the real numbers to the real numbers, including ones that are not always differentiable. For example, y equals absolute value of x is not differentiable at all points. But I would include that in the set F. Now I'm going to find a process called D, and I bet you can guess what it's going to be, that takes things in the set F D and sends them to F. And the process is simply the derivative. So I'm going to take D of little f, remember a little f would be an element. That would be a function that's an element of this set F of D. And I'm just going to return the derivative of f. Just take its derivative and that's the outcome of this process. So this is a process that is totally, totally familiar with you if you've had calculus one. But we can think of this process as a function. Or at least we think we can. Let's go through our usual questions and ask whether everything's in place. Did I tell you what the input set was? Yes, the input set here is a set of all differentiable functions from the real numbers to the real numbers. Is the output set specified? Yeah, that's the set of all functions period from the real numbers to themselves. Is the process specified? Yes, I said take its derivative. Now if you didn't know any calculus, that process would not necessarily be specified yet. I'd have to teach you what a derivative is. But if you know some calculus then, yeah, that just saying take the derivative of this function is enough directions to get the job done. Now the two important properties here, let's check those down. Does every valid input have an output? The answer here is yes, because I specify the input set to be all differentiable functions. Now if I had not made the distinction that I'm only considering differentiable functions for input, then the answer to this fourth bullet point will be no, because there are certain functions that don't have an output through this process. So if you're looking for non-examples of functions, that would be a good place to look, and we're going to talk about that in another video. Secondly, does every valid input have just one output? The answer there is yes. Every function you ever take derivative of has only one derivative to it. Now that's different if I were doing integrals, for example. We've seen or finding antiderivatives. We've seen if you take enough calculus, you see that there are plenty of functions. In fact, any function that has an antiderivative at all has more than one. But since we're going the other direction taking derivatives, yes, every valid input has only one derivative that comes out. So that's really a function. This d process is a function whose inputs and outputs themselves are functions. That's a fairly abstract idea, but it's actually pretty concrete too, pretty familiar. I mean you're used to thinking of derivatives as a process that you plug something into and get something out of. And here we're just making that sort of official. So now that this is a function, let's do a little review here and talk about the domain. Well the domain we already specified is the set of all differentiable functions. The co-domain is the set of all functions, period. Let's look at some images and pre-images. So for example, the image of the function cosine x, for example, would be negative sine x. And the image of x squared minus four would be 2x. So it's really easy to think about the images of these points under the function d, which is taking the derivative. And what about pre-images? For example, if I started with x squared minus four, which is a function, what could I plug into this derivative function to get it? Well that's simply a question about the anti-derivative. So this notion of finding pre-images is actually something fairly familiar to you if you've had a little bit of calculus. What is a function that I could put into here? Well, very simply just use your basic rules and you get x cubed over 3 minus 4x. That would be one pre-image. If I put that point, put that function into my derivative function, I get this as the output. But of course, there are many others as well. X cubed over 3 minus 4x plus 2 would be another pre-image for this particular point here. And of course, any constant you choose to add on here will give you another pre-image. We wouldn't say that there's only one pre-image with a plus c on it because the plus c actually isn't a constant. It's a whole collection of constants. So every point that has a pre-image at all is going to have many, many pre-images. Now notice there are some functions that do not have pre-images. I'm going to go over to this next slide here. What's the set of all actual outputs of this function? So remember we define the d function, the derivative function, to go from the set of all differentiable functions to the set of all just functions period. Now you may or may not realize this, but not every function out there has an anti-derivative. For example, the function y equals e to the negative x squared. That has no simple anti-derivative. That is, there is no function that I can plug in here and take its derivative that gives me e to the negative x squared. By no function, what I mean there is no simple function. And if a little bit more advanced mathematics, you can actually come up with a function. But I'm trying to keep it just in the realm of what you see in calculus here. So there's no basic function that when I take its derivative gives me e to the negative x squared. You may not have realized that that's the case, but it is the case. There are some functions that do not occur as actual outputs of this function. It's a little hard to specify exactly what the range of the derivative function actually is. So we'll just kind of say it in words. It's the set of all functions that have what we say elementary anti-derivatives. So it's the set of all functions in capital F such that little f has an anti-derivative. And that would be problem, that's a little unsatisfying, but it's probably the simplest way to express what the actual range of this function is. But do notice that the range is not equal to the co-domain here. That function is absolutely in the co-domain. That's a function from the real numbers to the real numbers. We can graph it. We could put numbers into it. It's all good. But that does not appear as anybody's derivative. And that can be proven using some pretty advanced mathematics here. So there's an example of a function that actually operates on functions. And I hope that blew your mind in all the right ways. Thanks for watching.