 Welcome back to another screencast on functions. We've looked at several examples of functions to flesh out and instantiate those definitions. And it wouldn't be a full instantiation if we didn't look at non-examples of functions. So you might be wondering after all these examples is everything a function? Or is it possible to create a process that somehow fails the definition of function? The answer is going to be absolutely yes. And this video is going to contain some of those non-examples. So just to remind yourself, there are five basic ingredients that go into being a well-formed function. We have to specify the objects that we're going to change that we now know as the domain of the function. We have to specify the process that we're going to undergo to make that change happen. That could be a formula or it could be a set of verbal directions. We have to have a set that holds the outcomes of our changes and that would be, we now know that as the co-domain of the function. And then we have these two properties at the end that every valid input needs to produce at least one outcome. And finally that any single input should produce only one outcome and not have any sort of splitting behavior like such. A function could, or a process could fail to be a function by failing any of these five criteria. So let's look at some examples of how each one might go down. For example, here are a couple of examples where we fail to meet the three basic parts where we either don't specify what the sets are involved in the function or we don't specify the rule. For example, if I just say that f of x equals radical x, that is not really a function. That might surprise you because that looks like something straight out of calculus or algebra. Well, in calculus and algebra, we have sort of unwritten rule that all functions go from the real numbers to the real numbers, okay? But if I don't say that, we now are in a situation where any kind of sets could possibly show up here. So if I don't actually tell you what the domain is or specify explicitly what the co-domain is, I really haven't given you a full description of the function yet. For example, maybe I mean the domain to be the set of integers or the co-domain to be the set of integers, okay? Well, if that's the case, this function has a lot of problems because not every, I'm not totally sure that this process actually sends things into the co-domain anymore. So in order to be a real function to be fully specified and well-defined, we must explicitly state what the domain is and what the co-domain is. So just stating the co-domain and the domain isn't enough, of course. For example, if I just say g maps the integers to the integers, that's obviously not a function because I don't know what g does. I haven't specified what is the process. There are a lot of ways I could send integers to integers, change one integer into another. What process do I mean this time? If I don't say so, then I haven't got a well-formed function. So a correct function needs to have the domain, co-domain, and process explicitly specified, otherwise we have more work to do. Now suppose you have the domain and co-domain and process all specified explicitly. You could still fail to be a function in a couple of ways. One way is that the process isn't defined for all points in the domain. Remember we said that every valid input from the domain needs to have an output, at least one output associated to it. And here's an example where I've specified the domain, the real numbers, the co-domain, and the process here. But this is not a function because not every point in the domain has an output. The point x equals two, which is definitely a member of the domain. F is not defined at two. F of two is not defined. We get division by zero there, and that is not defined. So if the function is not defined at every single point in the domain, then I don't have a real function on my hands here. Now it is possible in certain cases like this one to restate what the domain is. For example, I could say, well, that's actually the only point in the domain where I hit this particular problem here. And so I could restate the domain to say like, well, what I really meant was the real numbers minus the point two into the real numbers. Now if that's my new domain, then this is a perfectly legitimate function. Because every point in this set, which is a set of all real numbers except two, is definitely works. Every point in that newly restricted domain has an output. Now the final way we could fail to be a function is that you specify the domain and co-domain, you specify the process. Every point in the domain has an output, but every point in the domain might have multiple outputs. This is the dreaded splitting of inputs into multiple outputs problem that we have skirted up until now. But here's a couple of examples where we actually hit this problem. For example, let's define a function or a process, not a function called F, that takes the set of all classes at Grand Valley State University, such as Math 210, Section 1. And sends it to the set of all names of students at Grand Valley State University. And so the process says take a class X that would be like Math 210, Section 1, and return the first name of a student in class X. Now that is definitely not a well-defined function because although there's a process specified, and there's a domain and a co-domain, and the problem is that if I put in a particular class here, like Math 210, Section 1, whose name am I supposed to return here? There are 18 students in that particular class and I don't know which one to pick. Okay, so this is a situation where I have one input that splits into, in this case, 18 different outputs, a whole tree's worth of outputs here. So that is definitely what we would call a multi-valued function, not a function, a multi-valued process. So therefore it is not a function. One input splits into multiple outputs. Another more technical example is this one, defined the process int that goes from f to f. Here, f is the set of all functions from the real numbers to themselves. We saw this in an earlier screencast about calculus. And int of f is going to be an antiderivative of f. So the problem with this process is that it's multiple-valued. For example, int of x squared has not just one, but many different outputs. I could say x cubed over 3 is an output, but also x cubed over 3 plus 1 is an output. And x cubed over 3 minus radical 5 is an output. One input splits into multiple outputs. So that makes that a process, a process that we're really interested in. But it's not a function because it's multiple-valued. So there are five ways to fail to be a function. And just remember, when we're thinking about functions, all five of those basic ingredients do need to be satisfied. And we need to be careful about checking them. Thanks for watching all these screencasts and for all your patients.