 Welcome to our lecture series, Math 1050, College Algebra for Students at Southern Utah University. My name is Dr. Andrew Misildine, and I will be your professor in this lecture series. College Algebra, as the name suggests, is an algebra-based class, typically going beyond the standard topics taught in high schools, at least for US high schools, what have you. It has lots of different names though. Many people do take this type of mathematics class in high school, so College Algebra is sort of a silly name, but again, that's the conventional name that it's given. Sometimes a class like this might be called pre-calculus, because it's a prerequisite to calculus class. Many of the things that we will study in this class will be tools necessary to help us prepare for calculus. I often like to use the analogy that pre-calculus is kind of like the karate kid. I'm talking about the original one from the 80s, where you had this high school boy, Daniel, who's getting his butt kicked by the Cobra Kai all the time, because they're fighting over a girl, amongst other things. And so he's taken under the wing by this Japanese karate master, Mr. Miyagi. But at the beginning of Daniel's training, he's given these jobs to wax the car, paint the fence, sand the deck, and all these other tedious chores that are leaving him sore and bruised all the time, right? And so for Daniel, he was like, I'm just doing some grumpy old man's chores. I'm not looking, I'm not learning karate at all. But what we discover as we watch the movie here, Spoiler Alert, right, is that turns out that Mr. Miyagi was teaching Daniel Song these karate moves all the time, defensive karate techniques. He's trying to train his muscles to remember these basic defensive moves. And so the thing is, it looked like this tedious chore, but in the right context, it was the exact thing he needed to do to protect himself next time the Cobra Kai came around to try to beat him up, right? And so pre-calculus oftentimes feels the very same way. We're practicing very important algebraic and trigometric skills. But out of context, it's sometimes difficult to see why. And that's because the context is calculus, right? Most of these things we're learning is because it'll help us with calculus, our future mathematics settings, right? And so when you keep that in mind and you have that patience, we're trying to train ourselves for the tournament coming up in the next couple of weeks, you know, metaphorically speaking, of course. So that's one way of thinking about a class like this. Now, this class is not actually called pre-calculus. It's called called algebra, because we're only going to focus on the algebra side of that pre-calculus story, right? In order to get ready for calculus, you need good algebra skills and trigonometry skills. Now, this course will not prepare our trigonometry skills whatsoever. It's not that that's not important. It's just that's beyond the scope of this course. We will use a little bit of geometry as is relevant from time to time, but this will primarily be an algebraic course. And so because of that, it's sometimes called college algebra as opposed to pre-calculus. Back in the day, you know, when I remember today, you have grandpa sitting on his deck with his screen door and his hose ready to shoot all the kids in the street, right? He might have called a class like this FST, which is short for function, statistics, and trigonometry. Again, that just means pre-calculus stuff. We won't do any statistics. We won't do any trigonometry. This is a class that's just about functions. And so that's actually where we're going to start this course. In our first chapter, our first unit, we'll call it that, because these units don't necessarily correspond to any specific textbook here. Our first unit is going to be a unit on functions. We're going to learn the fundamentals about functions, and that's actually what's unit section 1.1 is about, this first lecture. It's about the fundamentals of functions, because our class, this college algebra class will be a functions class, of functions from a mathematical context. Now, when we talk about function, what we mean by mathematical function is we have some rule that assigns, well, we have two, let's start off, we have two sets. We have a set D and a set E. These sets are just collections of objects. Typically, these will be numbers, but these sets could be anything we want. D could be the people who are inside this class, D could be the set of all people watching this video, and E could be the set of colors. And our function could then be the rule that then assigns to everyone their favorite color. So, my favorite color would be green. You'll notice what marker I'm using right now. Your favorite color might be aqua maroon, if that's even a color, I doubt. But anyways, we could have a function that connects people to colors. That's fine. This is a mathematics class, so our functions for the most part will be sets of real numbers of some kind, our subsets of real numbers. So D itself will be some collection of numbers, E will be some collection of numbers. This number E, or the set E right here is typically called the domain of the function. This would be the set of all possible input. Some people like to think of a function as a machine that when you put something into it and it pops something out. The set E on the other hand is what's commonly referred to as the co-domain, right? These are the possible targets that you're shooting at. Now if you think of like a target in its usual sense, right? Like in a game of archery or something, you have like this. You're shooting your arrows over here at the targets, like this arrow. Cachink shoots that spot right there. You have another arrow that's, well, bam, shoots right there. I should probably use the different color there. The co-domain is the target space that we're shooting at. What we're gonna be more interested in is the range, where the range is the set of all values that we actually hit with our function. This will be the set of output. Now going back to the definition of the function itself, the function is a rule that assigns to each element X in the domain some element F of X in the co-domain, right? So if you think of this target example, my first arrow I shot, I got the outer ring. My second arrow I shot, it got a bullseye. And so let's say that the outer ring is worth, let's say two points, the inner ring is worth five points, then the next one's worth 10 points, and then the bullseye is worth 50 points. So what that means is I have a function where the first function, F of one, the first arrow gets evaluated at two because that's where it hit. And then the second arrow hits and gets 50 points because it was a bullseye, right? And maybe I gotta shoot three arrows. And so when I shoot the third arrow, I hit again the two-point range, right? And so F of three equals two. This is the idea of a function. And sometimes we draw these little diagrams one, two, three, and you have over here two, five, 10, 50. And so one goes to two, two goes to 50, and three goes to two as well. And so you sometimes see these diagrams to explain that the function is the rule of assignment. It decides which value in the domain, in the input range, comes over here to the output. Or we think of like this right here is our domain, and over here is this codomain. Code just means complement. It's the complement to the domain. And then the range, the range are those specific values that we actually hit. So the range would just be the numbers two and five. Now this course, like I said, it's gonna be about functions. And how much can we talk about functions? Apparently there's a lot we can talk about functions here. Now functions typically are represented in one of four ways, and all of these are very important things to be aware of. The four ways we typically like to represent functions is first, there's a verbal description, like we could describe the function using words. Like when we took this little story about the archer shooting arrows and such, I told you a story, but that story determined a function evaluation, all right? And so verbal descriptions, we as human beings, we tell each other stories, we talk to each other. And so we often describe functions relationships using verbal descriptions. Now in a math class, a verbal function typically means a story problem, which often gives people a little bit of concern. Sometimes we don't like story problems because they're challenging. And we sometimes feel foolish when we don't do things right. That's a thought process. I hope you will try to discard in when you're watching these lectures here and you're trying to learn mathematics. We make mistakes. That's fine. We learn when we make mistakes. We don't learn when we succeed. If you want to learn mathematics, prepare to make mistakes because that means you're learning something. So story problems will not be the thorn in our side for the end of days. But just so you're aware, we'll talk some more about story problems later on. For this lecture, the next couple lectures, we're going to stay away from them for a little bit longer. The next type of representation, we could call a visual representation. We can visually describe it. Now, one visual representation you saw with this function was this little table that was poorly drawn up here. This was one way of visualizing. This works well when you have not a large numbers in play here. I've had three arrows we shot. But we might also try to describe functions using some type of x or y axis. So we have the traditional Cartesian coordinates. And our function might do... I don't want to lie. We might have something like this as our function value. There can be some benefits of describing a function visually. It might be through a graph or something like that. Another type of function representation would be a numerical representation of some kind. Maybe we've gathered a lot of data. And then we know that this data represents some function relationship between two different types of quantities, two different types of variables. And so when you're talking about a numerical approach, you probably have like a scatter plot or a table or something like that. So you might have a... Like I said, you might have a table in which case you see something like this. And so let's see. We had x and f of x. 1 equals 2. 2 goes to 5. And then 3 goes to 2. So this is like the archery story we had a moment ago where we could actually record all of the function information inside of a table. This can be a useful way of describing the function. And then the last representation that we'll talk a lot about is the algebraic representation. So this would be something like f of x equals x squared plus 7. We can describe a function algebraically. Now, despite the fact this is a college algebra class, we actually are going to avoid doing this for a couple of lectures because as our focus right now is on functions, I don't want us to get too distracted with the algebra. We'll be doing plenty of algebra. Don't worry about that. But I want us to try to understand what are the important parts of being a function before we get lost or confused or slowed down by any algebraic skill sets. So we're actually going to do a lot of emphasis in the next couple of lectures on numerical approaches and visual approaches. So we can understand what's about the function and not necessarily the algebra. And I should mention, I guess before going on here, that one of the most important parts of college algebra or any other type of precalculus setting is that it's often useful to switch from one representation to another. You'll notice that archery example, I describe it in words. I drew a picture visually, but that picture was essentially just a table, right? So I kind of took three different ways of describing the same story. So we can try to understand what the function is. We gain new insights as we look at one representation versus another. Each representation has its strength, but they also have its weaknesses. And the ability to convert back and forth between different representations is the wax on, wax off principle that's a Karate Kid reference there that we are trying to gather here in college algebra. Now the first task that we're going to do with functions, you'll be asked to do things like identifying the domain or range of a function. Remember the domain is a set of all possible input. The range is the set of all, of all the actual output of the function. Another important thing we have to do with functions is evaluation. What does evaluation mean? It means that if I pick some x value, so if I could draw this as a table right here, if I pick some x value, we'll say x over here, what is the associated y coordinate, the associated f of x there? Like with our archery example, my first arrow was assigned two points, my second arrow was assigned 50 points, and my third arrow was assigned two points as well. And so this table is telling me that f of one was assigned to two, two was assigned to 50, and then three was assigned to two again. This is function evaluation. The different representations give us different ways of trying to evaluate the function. Be able to evaluate a function is critical. If you take any number in the domain of the function, we should be able to figure out what was the associated number connected to it in the range. Now of course, if you ask what about the fourth one, right? What's f of four? Well, it turns out that I didn't shoot four arrows, there was only three arrows in the contest. And so in this situation, we would say that f of four does not exist. Sometimes we write D and E for short. It was outside the domain, so there is no assignment for numbers outside the domain. On the other hand though, we're often tasked with solving equations, right? So when you do evaluations, when you do evaluations, you're given the x value, you're asked to find the y value, which in this case is going to be the f of x value. When you're solving an equation, what you're asked to do is you're actually going to be given the y coordinate y, and then you're supposed to find the x coordinate. So for example, we could ask something like, solve the equation f of x equals 50. Did you ever hit the bullseye? And in which case the solution would be, oh, x equals two. When x equals two, we got a bullseye with our arrow example. So solving the equation f of x equals 50 would give us x equals two. If we were to solve the equation f of x equals two, in that situation, we would get, oh, x equals one or three. There was multiple solutions, there was multiple times for which the function was equal to two. Now this is one thing that should be very clear when we think about functions, that with a function, every value in the input needs something associated to it. So one got assigned to, two got assigned 50, three got assigned two, that's okay. Every number in the domain gets assigned one. We can't be like, oh, arrow one got two or 10. It can't be both, right? It's got to be one or the other. Did arrow one get two points or did it get 10 points? Make a decision already. In which case they're associated to every x coordinate, there can only be one y coordinate. That's what makes it a function. There is a definite association between the number in the input with some number in the output. But it could be there are multiple output that are connected to different input. There could be two different input that produce the exact same outcome. So my first and third arrow both produced the number two. They got two points in the archery tournament. When it comes to solving equations, we could also do things like the following. You solve the equation f of x equals 10. Did I ever get a 10 point shot? In which case in that situation, we have no solution. It turns out that there never was a time when my arrow hit the 10 point range. And this is often abbreviated no solution where you draw a circle and they do a slash through it. This would mean that the solution set is empty. And so we can solve equations based upon the rule there. What could also work with inequalities, right? Inequalities will throw greater than or less than symbols into the mix, right? So can we solve the situation when f of x is greater than or equal to 10? Well, based upon this function, we'd be like, oh, if f of x is great equal to 10 when x equals two, because in that case, f of x was 50. Could we answer the question, when is f of x less than five? Well, that happened when x equals one or three, just as some examples, right? We just want to figure out, when is the y coordinate greater than equal to 10 or less than equal to 50? These are the type of questions we're going to approach here. And the way that we approach them will depend on the representation. A tabular function, we might solve the equations differently than we would a graphical function or an algebraic function, which we'll see some of those in the next coming videos. Make sure you click the link right now. You click the link you can see right now and you can start the next video and learn how to solve equations and such for these different functions and different representations.