Hi, and welcome to math for data science Course 1, where we're going to be talking about functions and rates of change. Today we're going to talk about a review of functions and algebra to get you going in the math that you need to be successful in data science. I'm going to start with just a review of what is a function. When I think of a function, I think of a box that's going to take some input, and in this case, I'm just going to call that input something crazy like stick man, stick person, and for a given input, it's going to do something to it inside of this box and it's going to give me an output. Let's just say, for instance, maybe this particular function box is going to take whatever I gave it, square it, and add one. I'm going to call this box f and I will describe f by f of whatever I want to say. I could call it x or I could call it stick man, I'm going to stick with stick man for the moment. The f of stick man is stick man squared plus 1. If I put an x into this box, then it would give me out x squared plus 1. If I put a zero into this box, it would give me 0 squared plus 1 or 1. If I put a negative 1 into this box, it would take negative 1 and square it and add one, which would be 1 plus 1, or 2. If I put a positive 1 into it, it would square that and add one, so it would give me 1 plus 1, or 2. You see for anything I give in, it's going to give me exactly one answer out as my output. Usually, it's going to be described this way, usually, we will write it as f of x equals, but I don't want you to think about that being like x, it's of whatever the input is, is what I'm going to do to it. When I want to think about this, this is the way to describe it algebraically, but if I want to describe it graphically, then I could create a graph. I'm just going to put the graph over here, and let's say this x-axis be my input and my height or y is going to be the output. I will put some points in. When I did 0 for my x, I got a height of 1, so that is there. When I did a negative 1 for my input, I got a height of 2, and I did 1 for my input, I got a height of 2. Let's see, if I did two for an input, I would get 2 squared plus 1 or 5. Let's just put that point, 3, 4, 5 is going to be about there. If I put in more points and connect them with a line, I get a graph that looks like that. I can see that is the graph of a function. Because for any x value that I give, there's only one possible value here. Now that doesn't mean I can't get the same answer twice, because here I got the same answer in two different places. For two different inputs, so if I gave it 1, I get to the same height as if I gave it a negative 1. That is still a function and a valid graph of a function. What kinds of things would not be a function? From a graphical perspective, for instance, if I turned this graph on its side, something like this, that's not the graph of a function because what happens when I give it 0? It goes, well, I don't know, it might be 3, it might be negative 3. It doesn't know what answer to give you. Think about the box. If you gave it a value and it gave you two possible answers, then it's not a function. That's how we define a function, is it operates on an input and gives you a result, that's one result for the input that you give it. The result could be repeated, but it can't be indeterminate, like it doesn't know what to give you. That's an overview of what functions are and how they work. We're going to give you a chance to practice that just a little bit and then we will come back and do some basic algebra review for you before we move on. Thanks.