Hi everyone, and welcome to our course. We're going to start off by studying functions, functions and how they're relate particularly to models. The fundamental objects in this course that we're going to study in the next courses to follow will be functions. What we're going to see here, this is going to prepare the way for future study of calculus. We're going to discuss the basic ideas concerning functions. We're going to talk about their graphs. We're going to talk how to transform them. And we're going to combine them as well. And we are going to show the many different ways that you can see a function and how to know one when you're looking at one. You can see with equations or tables graphs even in words. So we're going to look at all these things. We're going to look at the main ones that occur in calculus, describe the process of using these functions as mathematical models of real world phenomena, and look at them both in a single variable and multivariable case, okay? So I would imagine most of us at some part have seen what a function is. So let's define a function. We're going to say that a function f is a rule. This is a very abstract definition here. A rule that assigns to each element in a set D exactly one element called f(x) in a set E. So our sets for functions here is these are going to be real numbers specifically in this course. We're going to talk about rules that assign. The key here is that it's exactly one element and I'm sure we've seen lots of functions. But in particular, sometimes they draw this picture where it's some machine where you feed it, feed it a number into the function. And it will output some other number. This is usually the stereotypical drawing that you'll see everywhere. And you've probably come across lots and lots of functions. Another picture that you might see for functions. Sometimes they draw a big circle or oval or whatever, and they represent dots. This set over here, your input, sometimes called the domain. These are the set of numbers that you are allowed to plug into the function. And you can call them little a, little b, little c, whatever you want. And the function will output them to some other numbers. Maybe the first one goes to f(a) and the second one goes to some other numbers. So they're just a rule that assigns one number to another or a bunch of numbers to another one. We're going to start off in this course studying functions of a single variable. This is important, and you're going to usually see these things as when one number comes in, then one number comes out. This is how we're going to start off with our study of functions. So let's get away from the abstract definition. Let's get actually to some specific examples and ways to represent functions. A lot of the times, the thing I think most people are most familiar with is you can see them algebraically. This is where I give you a function. I'll say f(x) is this is the rule, I'll say x squared. Or maybe I'll say f(x) = 2 x + 7. Think of this as a game. I give you a number, you do something to it. So for example, the first one, f(x) = x squared, I give you 2, you give me 4. I give you 3, you give me 9. Whatever the rule says, that's what you do, it's a robot. You don't really question it. You just say, okay, give me whatever it takes. The second one, 2 x + 7, you give me 1, I multiply it by 2, and I add 7. Give me 10, I multiply by 2, and I add 7. Whatever it is that you give me, this is what I will do with the rule. You can also do things, of course. Visually, you can look at things and say, well, what if I just handed you a graph. A lot of people are very familiar with this. The x squared graph is usually the one most people are familiar with. You can draw it and pick some points. So at 1, you can have 1, and I can hand you may be a data point at 2. At 2, we look at 4 maybe, picture not drawn to scale, and you can go up and up and up and up. And if you ever worked with any data, you've seen scatter plots and you can hand you a bunch of data in a table or a spreadsheet. These are other ways to sort of represent functions. I will hand this to you without a rule. Think of any spreadsheet that you've ever seen, most likely it's a column or tables of data. There's not necessarily a rule behind all of that. We can also do things with graphs. Let me give an example of a graph. This is from the Meteorological Service of Canada. This is a graph of the weather and you can see that it's color-coded, this is a nice visual graph. It doesn't have to be on the xy-plane that most people are familiar with. All of these things, how every model is working, this is a function. So any time you see a fun graph, 2D, 3D, whatever is representing their sum function that is behind the model. This is probably more like one that you're used to here, I have an xy-plane. I have a graph that's kind of going up and down on the graph. And you see these things and you say to yourself, okay, what are some important things I want to know? What are some values that I care about? Well, we can look at, remember, we have the x-intercepts. The x-intercepts, where does the function cross the x-axis from the graph? We have a point at (-3,0). There's another value here at the origin (0,0). So you can certainly have more than one x-intercept. And we have a nice (0.9,0) of the graph, x-intercept at (1,0). Sometimes these are called the zeroes of the graph. They are the values or the points, so the zeroes would be the x-coordinates of them. So -3, 0, and 1, this is where the function outputs 0. And yes, the zero is a 0 of this graph. There's other points you can talk about. You have a nice little maximum on this graph. You have a nice little minimum value here. And we have a nice pretty graph here. Notice there's no formula given. That's okay, doesn't mean I can't look up some values and guess or estimate what some values are based on the picture of this graph. This is just another way to present a lot of information. And the idea is every function just represents a transfer of information, some rule that's being determined on how these functions behave. A lot of times you'd like to know when you're looking at a function. So lots and lots of things are function, but unfortunately, not everything is a function. There's something called the vertical line test that'll help you when you have a visual representation of the graph to determine if this is in fact a graph. Informally, this is how this works. I'll hand you some graph, let me give you something like this, and I'll say to you, hey, is this a function? Is this the graph of a function? What you can do in your mind or if you have pencil you can do it, you draw a bunch of vertical lines. Remember vertical lines, they go up and down. Draw a bunch of vertical lines anywhere you want and try to draw a line that crosses the graph twice. Okay, so if a vertical line, this is what the test says, if a vertical line crosses the graph more than once, then it is not the graph of a function. In the picture that I drew here, I do a bunch of vertical lines. It's impossible to draw a single vertical line that will cross this graph more than once. So this is in fact the graph of a function. However, you can imagine there's lots of pretty objects in math that you've seen before. So maybe like the circle, for example, if I draw the circle and I say to you, hey, is this the graph of a function? You come along with your pencil and you start drawing a vertical line. And pretty quickly, you hit the graph in more than one spot, the top of the circle and the bottom of the circle. So this would not be the graph of a function. Whereas the first one is a graph of a function. And the circle is of course not the only object or only graph you can see that would fail the vertical line test and not be the graph of a function. You can certainly have graphs that maybe double back on themselves. So you have a graph the moves to the right and all of a sudden moves left, that'll be bad. And all it takes is one vertical line, one vertical line, boop, done, that hits this graph in more than one spot and the it's an automatic fail. You need every single vertical line that you could possibly draw two always cross this graph just once. You say, well, isn't everything important a function? Can I just ignore these things? Well, then you'd be thrown out the circle, that seems like a bad idea on math as well. And here's another example I grabbed from the National Hurricane Center, some 2010 Atlantic hurricane season, some tropical cyclone tracks. And you can follow along on this map and see that some graphs, hurricanes, do go back on themselves. So to study and model natural behavior not only needs some functions, but you have to work with things that are not always functions. And we'll develop that theory inside this course. So it's pretty natural to have things that are functions and pretty natural to have things that are not functions as well. All right, let's do some examples of functions, some functions that we like to some easy ones just to get started with. These are ones that, certainly, think of it as building your mathematical vocabulary that you can play around with. The one we saw before just to get started would be f(x) = x squared, that's a nice one. It's nice and symmetric, as xy-axis symmetry. So if I draw this thing, we call its graph a parabola. It looks like a letter U, this goes right through the origin, and you can do it this way. So that's a nice little graph. And the idea is to build your intuition. You want to know both what the formula is and then some have some behavior, some understanding of what the graph does if you were to sketch it out, what the general behavior of it is. Another one that I think other people have been familiar with you might remember is something like a line. So we can do, let's start off with an easy line, f(x) = x, and they call this the identity function. Whatever number you give me, I just give it right back to you. So you give me 1, I give you 1. You give me 2, I give you 2. Work this out, you draw this. You get lots, you get a nice straight line with slope 1 that goes right through the origin. And again try to think of some functions that you may have seen or remember in the past, keeping kind of simple. What's another other nice one that we can play around with? How about the function f(x) = x cubed? Sometimes they call this the disco function, move your hand in the shape of the graph and you'll see why. So the the function x cubed starts off from low and it goes to high. It goes right through the origin that's sort of a nice one there, and one more just to make it a little more challenging, I guess. Let's do the square root graph, the square root of a number. This is not defined for negative numbers. You can only input 0, the square root of 0 would be 0, start at the origin and then off you go. It gets larger and larger and larger, and it's a little different. All these have their own characteristics, and we'll go through some of these characteristics and talk about them. But here's just a couple that you might see, there are other ones. If you ever get stuck, you can throw them in technology on web pages, in a calculator, and graph. But for some, you want to have a nice foundation of what the functions are, how they behave. And I always compare it like if you have a conversation with someone, you don't want to run to the dictionary to look up every word that the person's saying. Same thing, if someone's talking to you about something with exponential growth or linear growth or parabola, you should have in your mind the general shapes. It doesn't need to be perfectly exact, but you like the general shapes of what these things are. In addition to functions, we have what's called multivariable functions. So before, before, before, before, we had a function that took in one number and output one number. These are called single variable functions because there's one number in. Let's talk about the descriptor's always on the input. So these are single variable functions. And now what I want to introduce to you is something called a multivariable function. This is the idea, well, what if I hand in two pieces of information and assigned a rule that works with that? So for example, how about something like x + y? So you give me two numbers and I'll just add them. So I hand you 2 and 7, you throw in the machine, you throw into the function and it says 9. This is what's called a multivariable function. There's nothing stopping you from throwing in more. You can certainly have three numbers that come in or four or five or whatever and maybe we do x- y + 2z. Whatever the rule is, you just follow the rules. So I'll hand you three numbers, and you take the first one, subtract the second one, and then add 2 times the third one. And you can plug these in all day and evaluate these things. You can graph these as well. I have a picture here of, if you see, this is a cone. It's in 3D, their graphs. They live in, they're not just flat. They have some sort of volume to them. They live in a larger place than just the xy-plane. The graph of this cone if you notice, the center at the origin, it is of the equation f(x,y) = square root of x squared + y squared. All the good websites that graph things in two dimensions, that single variable functions, they will also graph for you three-dimensional functions. The graph here in this, I use GeoGebra to graph this. And that'll be some exercises where you go through and just graph these things and find pretty pictures. This is used a lot in modeling. It's kind of fun to look around the room and see what 3D objects you have, and think about what functions were used to program it to some machine to make those objects as well. But when you see functions, you should understand that they're described by their input. So single variable in, means you're a single variable function. More than one number in, two three, four, whatever, that's a multivariable function. The input describes what they are. So let's look at some more examples of multivariable functions. These might be new to some folks who have not seen these before. So let's just start off with just the simplest kind of multivariable function, a function that takes in two things. And let's look at x squared + y squared. So a couple things I could ask you, same thing I would ask for any other function, how do you evaluate this function? So for example, if I said let's plug in 0,0, what do you do? We just follow the rule. This would be 0 squared + 0 squared, of course, that's just 0. If I say, well, what's f(1,2)? Well, that says take the first component, square it, and then square the second one, that's 1 + 4 and of course, that's 5. So like any other function, it's a little bit of a game. You give me a couple numbers, and I give you back, then you graph this. What's a little more interesting if you can try to get away from just listing numbers and think what would the graph of this look like, it's two variables in. So it has two variables that you would have to give it which means that its domain or set of inputs is the xy-plane. When we draw things in 3D, the default picture we draw has three axes, two for the input, x is coming at you, the bottom left, y goes to the right. And then your z variable, your output variable, goes up. When you graph this in technology, you can of course graph this thing and spin it and get the most perfect view. But if you're just trying to graph something to get a handle of it, this is the typical use of the axes you draw, the x, y, and z axes from multivariable function. And you can pick your points. We already saw that we have 0 when I plug in 0,0 on the xy-plane, the output is 0. And if I plug in 1 on the x-axis and 2 on the y-axis, so 1,2, that lives out here a little bit. Then I plug in and I get 5, so you go up, [SOUND] to get 5 on the z-axis. Now this would be terrible to draw by hand. So the challenge here is to sort of pause the video, go off and graph this thing and see what you get. It turns out that when you graph this thing, x squared + y squared, you can never have an output of a negative number. And think about that for a minute, right, take any number in squared and then the other number in squared, positive, 0, negative. Squaring always gives you a positive number. When you add two positive numbers together, you get back once again another positive number. So the graph of this function is always going to live above in the xy-plane. There's nothing going to be down here below the floor. And it turns out there's a lot of symmetry going on here, it's like parabolas. When you have this thing, you get a parabola in the xy-axis. You get another one and you get this bowl-looking shape. I'll try to do my best to draw it here. But I'm going to recommend you go using technology. You get a bowl-looking shape and its got nice, I guess, circles if you're looking straight down at this thing. And you can get this bowl shape to this thing. I won't ask you to graph these things or develop but sometimes some of them are basic objects. Again, this is usually very difficult and you can only kind of visualize this in your head or by hand for simple examples. But knowing more importantly how to graph these things if you had to get a picture, some intuition of what's happening for these multivariable functions, that's almost more important there. Okay, so we've seen basic examples of functions. So the vertical line test what is and what is not a function and looked at single and multivariable functions just as a starter. We're obviously going to study these things more throughout this course, so great job on this first lecture, and we'll see you next time.
