 Welcome to the screencast on functions. This is a really important video, because functions are probably one of the most central concepts in all mathematics. You've already dealt with them quite extensively if you took calculus. What we'd like to do here is expand our idea of a function to include not only what you've seen in calculus, but also lots of other things as well. So what's the big deal about functions? Well, in mathematics one of the main ideas we think about is change. How can we change one quantity into another? Is it possible to change one quantity into another? In calculus you ask lots of questions like this about change. At what rate is a quantity changing with respect to time, or with an integral you ask how much change has accumulated in a quantity over a certain amount of time. This line of questioning continues throughout further subjects like linear algebra, where we ask things like if we stretch the x, y plane out a certain amount and then reflect it over a line, are there any vectors that don't change their overall direction? We care about change because the world changes and we want to be precise about the way in which change occurs. Functions are the tool we use to model and understand change. And functions really consist of three parts. First, they're the things that we intend to change. We usually think of those things as belonging to a set. Second, there is a rule that we follow that tells us what to do to those things to change them. And then third, there are the things that the objects in part one change into. Those also belong to a set, possibly a different one. So here's a simple example using numbers. For my set of things to change, I will let that just be the entire set of real numbers. My rule I will follow is given a real number, I want to round it up to the next higher integer. For example, if my real number that I start with is 5.2, then my rule says to turn the 5.2 into a 6, the next higher integer up. If my real number is pi that I start with, then this is going to get changed into 4. If I start with a 10, then that's already an integer, so the result is just 10. If I start with negative 3.8, then the next higher integer up from there is negative 3. Remember that negative 3 is greater than negative 3.8. And this rule applies to any real number. The set that contains the outcomes here is really the set of integers. So this is a different set than the one that contained the things that we put into the process. Now note that there's two important things about this process, too. First of all, every real number can be run through this process. There is no real number that cannot be rounded. And secondly, and very importantly, when I take a real number and run it through this process, the outcome is unique. That is, it's not possible to take a single real number, run it through the process, and end up with two different results. A real number will only round up to one thing, not two. Those two properties are very important for having the kinds of processes that we want. Now, here's another process that's very similar in how it works, but it's not something you'd normally see in a math class, and that's a vending machine. A vending machine is basically a large piece of hardware that implements the change process. How does a vending machine work? Well, you walk up to it, you put in a certain amount of money and punch in a code, and the machine goes through and dispenses your food. How is this like our numerical process from the earlier example? Well, the things we intend to change here is a combination of money and text. You have to feed the vending machine both a certain amount of money and a code. The process is whatever the machine is programmed to do. It will read the amount of money, read the code, and then send the little robot arm or whatever inside the vending machine to get your food to the right place. And the outcome, that would be just the food that you get. So notice that the vending machine has these two important properties as well. Any valid combination of money and vending machine codes will, if the machine is working properly, return something. And secondly, if I put in a valid amount of money and a code into the machine, then the outcome is unique as much as I would like to be otherwise. Only one food item is going to drop out if I give it a certain amount of money and a code that's valid. So here's a third conceptual model to help us understand how functions will eventually work. And here at Grand Valley State University, every faculty member and student has an eight digit identification number called a G number. Quite often, for example, in the admissions department, we want to change names into G numbers. For example, if I want to look up one of my advisee schedules, I have to enter in their G numbers into a special website and not their names. But I know their names, but I have no idea what their G numbers are. So I have to have a process for changing a name into a G number. And that can be done via a special website that advisers can access. So let's run through and see if this has the three basic properties of any process we want to consider. The things that I intend to change here will be the set of all names of faculty and students at GVSU. The process of change is the computer program that's on our website that takes this first and last name combination and does a database look up and then prints the G number to my screen. And then the outcomes of the change would be the set of integers between zero and nine, nine, nine, nine, nine, nine. So notice again, with this process too, those two important properties hold. Any valid combination of first and last name, if put into this website properly, will produce a G number. And if I put in someone's first and last name, that person will have only one G number. It would be kind of bad if one person had two identification numbers. So the outcome of the query there is unique. So now we're ready to formally define what a function is and we'll do that in the next video. But for now, let's take away the things that all three of these conceptual models have in common. There are five things to note. First of all, for each of these conceptual models of a function, there's a collection or a set of objects that we are putting into our process. These are the things that we wish to change. Secondly, there is a process that carries out that change, or at least describes how we're supposed to do it. Thirdly, there is another set that holds the outcomes of that change. And then fourth and fifth are two important properties. Fourth, any valid input to the process needs to produce an outcome. And finally, any single valid input to the process produces only one outcome and not two or more. So those five ingredients are the keys to defining a robust notion of a function that will stand up to the use to which we will put it. Thanks for watching and stay tuned.