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Functions on the Real Line - Overview

In mathematics, a function (or map) f from a set X to a set Y is a rule which assigns to each element x of X a unique element y of Y, the value of f at x, such that the following conditions are met: 1) For every x in X there is exactly one y in Y, the value of f at x; 2) If x and y are in X, then f(x) = y; 3) If x and y are in X, then f(x) = f(y) implies x = y; 4) For every x in X, there exists a y in Y such that f(x) = y.


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Video Transcript

So I said that the hardest part of calculus is algebra. The other hardest part is probably notation and just getting a real sense of going from okay, just solving equations to really understanding how to communicate in the math language. And so we're going to do just a very brief introduction to set notation because we're gonna move into talking about functions defined on sets. And so the first thing we have is the natural numbers. You'll see this little blackboard in is the symbol. So that's just the counting numbers. 1234 etcetera. You have the imagers that's denoted by the blackboard, see, And we include zero and all of the negative numbers. Then we have the rational numbers denoted with blackboard que and these air basically fractions. But they're just questions of integers where, of course, the denominator cannot be zero. We can divide by zero, and then what we're really interested in is the real numbers. That's what calculus is really going to focus on, is gonna focus on the study of functions defined on the real numbers. And so what are the real numbers? Well, it's pretty easy to follow how to define the rational numbers. If you believe, uh, there these things called the counting numbers that allow us to enumerates sets and then the integers you just sort of generalized you had in zero, you know, which was a big deal. You had a negative numbers, which was also a big deal. And then you just take questions of those to get the rational numbers. But what are the real numbers? Well, the real numbers are just the rational numbers with all of the holes filled in. And I'm not going to spend a lot of time saying much about that. If you take a class in advanced calculus or analysis is what it's called, then you'll talk a lot more about how the real numbers are basically defined from the rational numbers. And so I guess the only thing I'll say is, What do I mean here by hole? Well, it turns out that there holtz in the rational numbers. So in other words, if I think about well, is there a real number X such that when I square it So I multiplied by itself? I get to Well, it turns out that X is not rational, so if I define this thing called Let's just say the positive square to two. The number that when I squared I get to this X is not going to be a rational number, which is kind of a deep problem. You know, somehow you might think that all numbers irrational, but it turns out that their numbers that are not rational and so those air kind of the holes that exist within the rational numbers and and that's really all I want to say about that. But what I do want to connect up is, once we have this idea of the real numbers, calculus is pretty easy to nail down. So calculus is study of functions defined on the real line or, more specifically, on subsets of the rial line. What we'll do next is will actually be a little bit more specific about what types of subsets we're going to focus in on. So when we get into different types of functions, we're going to start throwing out domains of those functions, and we're going to get a lot of things that look like this. Closed intervals, open intervals. This is kind of halfway a joke club and intervals so closed and open intervals closed, raise open, raise. And then we can combine intervals with these operations unions and intersections, and I'll just leave This here is a reference you can look it over and try to make the connections with everything. I hope that you've seen this before, but we just want to review the main idea. So notice, You know, first of all, notation Aly that when we use a square bracket, we're including that in point. When we use parentheses, we're not Including that in point. Uh, on the right hand side here, I'm using this set builder notation. So we'll see this time and time again, especially if you go into any more advanced math classes. It comes up a lot more, but the idea here is that if I write okay, so I'm saying I want a real number x two and X. That's in the real numbers such that some property of X is true. So you see, here the property is that X lies between these two numbers A and B, but in general it's just saying I want a real number, such that some property, that real number is true and then that defines us that now the set might be empty. There may be no numbers X that satisfy this property, but that's the idea behind set builder notation. And we'll see it here as so So the last thing I'll mention is that actually weaken, right. The entire real line in interval notation is just minus infinity to infinity. And, you know, there's also these cool diagrams you can draw. So, for instance, if I have this interval here, then Aiken plot this on the number line like so So here in zero here would be one. And so I can plotted by putting a solid 0.0 a solid dot at one and connecting them to show that Okay, this interval 01 is plotted like that. Okay, there's similar waste. A plot, all these other types of of intervals. For instance, if I wanted to plot this ray to do infinity where I don't include two Well, then what I could do because I started to and I draw a open circle to signify that. Okay, I'm not including two. This is too. And then I can just extend off with an arrow this way, showing you know im, including every real number greater than to, but not including two. So we use this idea again, and we'll actually generalize that idea later. When we look at Functions way, start drawing the graphs of functions in the X Y plane. So the next thing I want to dio is introduced a very important operation, and later we'll see it's actually a function, but that's the absolute value of a real number. So have a real number X. The absolute value is defined like this. It's defined, actually is a piece wise function, so it's just going to be itself. So in other words, the absolute value of the rial number will be itself if that number is greater than or equal to zero. But then, if it's less than zero or negative, I'm just going to say that the absolute value of a number is negative. So if it is negative, it will become positive. And this is also a great chance to introduce the piece wise notation that I'm sure you're familiar with. So for different parts of the domain of dysfunction or for different real numbers, it has a different definition, and we're going to see this time and time again. So again, lock this in your mind because it's really important. But notice that there's another way to define the absolute value of a number, and that's to take the square root of the number square and notice what happens here. If a number is positive and I square it and then square root it, I just get the number back. But if the numbers negative, I square it, I get it's positive square, and when I take the positive square root I get, it's absolute value. All right, cool. So this is going to come up a lot of the absolute value when solving equations and inequalities. So the thing to consider is that if I have a variable X, and I know that the absolute value of X is equal to some real number, a, then one of two things could be the case. Either excess a or minus X is equal to a because I have to consider both parts of the definition. Either ex congee A or the negative of X can be a and so that's what this equation means. One of these two things can be true to make the opposite value of X equal today. All right, So now that we're getting a handle on the idea of sets and subsets specifically of the rial line, let's just go ahead and reintroduce the idea of the function. And this is going to seem a little bit abstract at first. But it's really important to kind of get this idea in your head. So a function which will note for now just by a little F Sometimes we use different letters, but a function f from a set de to set why and now these sets D and why we really want to think about them is subsets of the real line. So maybe an interval, maybe a union of two intervals, something like that. And so to give the definition, I'm just going to draw a picture. So we're going to start with our set D, and I'm just going to draw it as a well, not really a circle but region. So here's D, and here's why. And so what it does is it takes something in D. So let's say X is something in D, and what it's going to do is it's going to take it over into something in walking and we call this something And why? Well, little why like that? Okay, and this is the function. So the function is going to take everything in this set D in a sign, each thing indeed to exactly one thing and why and now that's really important. That's if you recall. That's like passing the vertical line test. So for every input of the function, we get exactly one output. And now the said D is called the domain of F. In this set, why is sometimes called the co domain? That where it isn't as Communists domain? That's what it's called. And so what we all right is that this element why in this in the code of Maine is a function, or is an output of this input X in the domain? And when we use why an X for these variables will say that X is the independent variable for the function. And why is the dependent variable? Because it depends on which X we choose for an input, but we're sort of free to independently choose X from the demand. But then we once we picks ex, were sort of fixed into which output we get in white. So that's the basic idea of functions. And just to kind of refresh your memory will next give some examples of why functions or useful. So here, just a couple of examples are actually specifically three examples of different types of functions you may encounter. And so let's just identify what the independent and the dependent variables are. So in the first example, the position of a ball, which is given by the dependent variable s is a function of the independent variable t. So for each time after the ball was kicked, I plug in t into my input and I get my output, which is a position of the ball. Same thing here. The deepening variable here is H the height of the plant, and it depends on the independent variable D, which is the days after it was planning. So I have input output. Same thing here, Deepen it. Variable is pressure, Pete, and it depends on the independent variable T, which is the temperature outside. So I plug in temperature outside, and that should give me ah, unique pressure in my car's tires. So there's a few examples, and I'm sure you can think of a lot of examples on your own. So I've talked about the domain and the co domain of a function. But in the calculus setting and in a lot of settings, we're not explicitly given. A Domain and ako domain for a function were just given, ah, function were given that Why is some function of X? Maybe this is X squared sine of X. Either the X. We're going to talk a lot about those examples later. But let's just say for now we have a function and we have a dependent variable. Why independent variable X. We're going to define two very important sense in the first one is called the natural domain of F, and now notice. This is a little bit different than the domain of that, because I'm not actually going to specify what the domain is. I'm calling it the natural domain because it's exactly that. It's basically all of the rial numbers X, for which ffx is a real number. So let's actually write that in set builder notation so well denote it by D of F. And what this is going to be is it's going to be all the real numbers such that ffx is a real number. So think about, for instance, if the function was one over X, well, we can't divide by zero so X would be excluded from the domain of one over X because 1/0 is not a real number. So we're just looking at the numbers, the real numbers that we can naturally plug in to the function itself. The next concept is one that I'm sure you're familiar with. And that's the range of the function f. And so the range of F, which will denote are a Beth is gonna be the set of real numbers. Why, through which, what? Well, for which there is and X in the domain of f such that why is equal to ethics. So in other words, we're just looking at all of the real numbers that air out, put it by the function. So you you look at all possible inputs and then you just see what happens when I plug in those inputs into my function. The collection of all possible outputs is the range of F. So here we have a bunch of different ways that we can combine functions. And again, I'm sure a lot of these air familiar to you. But I just want to put him here just as a reference and just to review and notice that I've also included for each one of these ways to combine functions, the definition and also what the domain is going to be. So, for instance, if I have a function F, I can take a constant multiple of that function. So you just multiply the function by that constant multiple, and that's just gonna multiply all the outputs. By that, whatever that real number is C and that's not going to change the domain of F. And then we can add two functions. We'll see that the domain is just going to be the intersection of the two functions. The intersection of the domain of the two functions product of two functions, is the same thing. You just take the product of the outputs and the domain is like I said, it's the same. It's the intersection. Question is just like product. But we just need to add this extra restriction that the function that sends a denominator can never be zero And then finally the composition of functions is just going to be well. You take the input of one function as the output of the other, and the domain is going to be basically wherever that makes sense. Eso Here it is in and set notation. You could look at that, but basically you just want okay, I want the output of G to be in the domain of F. Otherwise it's not going toe make sense to compose the functions. Another thing to notice about composition of functions is that it is in general, not community, meaning that if I take f of g of X, that in general is not going to be the same thing as G of F FX, and that's gonna be important later on. We're going to see some examples that really throw some curve balls and maybe the way you think about functions. So I just mentioned that function composition in general is not community. So, for instance, let's suppose that ffx is equal to X squared and g of X is equal to square root of X. Okay, so let's think about what is f of g of X, well, G just tells me to take the square root of the real number, and F tells me to square it. So what this is going to be is it's going to be the square root of X squared, which is X with one condition. Notice that we first have to take the square root of the number. So that means the domain of this composition is going to be the same is the domain of Jeep, meaning that the domain is not going to include any negative numbers. So the domain here is going to be X greater than or equal to zero. So this isn't really the function X where I could just plug anything yet. It has a natural domain restriction. So what about G off athletics? Well, this is going to be the square root of X squared. And now what is this? This is something that we've already seen. This is absolute value of X. Now, the domain of this function is going to be all real numbers. But actually, the output or the range is what's interesting. We're not going to get any negative numbers. We're only going to get non negative numbers. And so you see, that these two functions are different. They have different natural domains because I can't plug in a negative number here, and I can plug in a negative number here. So you see that the function composition is not the same, and this is an extremely important example. And I would really encourage you if you're saying what is going on right now to really sit down and think about this example, because it illustrates how intricate functions can be. And when you start composing functions, how careful you need to be later on, we're going to see taking the square root of something squared. We're gonna have to think about that as being the absolute value. Andi and it's really interesting and it's really worth spending some time thinking about. But we actually want to think about the case where dysfunction composition is community. It does go both ways, and actually in the special case where F of G A backs equals G of Alfa Becks equals X, then we actually give these functions a special name. F N G are said to be in verses and a lot of times will write the inverse of F in a special way. So the inverse of a function f is denoted Yeah, to the negative one that members. And so another thing to note about the domain and range of inverse functions is this. Well, the domain of F is gonna be the range of f inverse. That's really nice to know in the range of F is going to be the domain of F members. And this really makes sense if you think about it, because to be in vertebral, if I start in the domain of F and I apply f I end up in the range of death, Well, that better be in the domain of Beth members, because I need to be able to go back into the domain of death to sort of reverse the process. So again, function in versus is something that's gonna play an important role and calculus. And so it's gonna be really important to understand. Maybe the simplest example of two functions that do not have inverse is X squared and square root of X. So this leads us to a really natural question. And the question is, this does every function have an inverse? And the answer, of course, is no. And we've actually already seen an example of a function that does not have an inverse on its natural domain. And that function is X squared. And the reason it doesn't have an inverse is because, let's say I started with the number two and I squared it when I went to go in square root or reverse the process. I have to make a choice. Do I mean to or do I mean minus two? There are actually two square roots of four, and so, in a sense, you cannot put the toothpaste back in the two for every function. So then the next natural question to ask is, When does the function having members? And if you recall to be a function for every input, you had to have exactly one output. So if we want a chance to reverse the process, we better have for each output exactly one input. And so, in other words, we want the function to be one toe one. So the answer is F R function has an inverse if it is okay once one. And so what does 11 mean? Well, it's exactly what I just said for each output. There's exactly one input, and we can write this symbolically. So suppose that I had f of X equaling f of Z. So I had to numbers X and Z that gave me the same out. But well, then it better be the case. If this happens, that X is actually just equal to Z. So it's not possible. Toe have two different inputs giving me the same output. And you can see that when this is the case, the inverse function is actually well defined. It actually is itself a function because, remember, my outputs become my inputs for an inverse function. And so if my output is here, I better have one input back in the domain of the original function. So now that we know when a function has an inverse, let's actually go back to our example of a function that does not have an inverse on its natural domain. And that's X squared has no in verse and another way to say this that will say sometimes so it doesn't have a members. In other words, it's not in veritable and specifically it's not in vertebral on its natural domain. But what would happen if I just restricted the domain. So I know that, too, and negative two are going to give me the same output when I square them. But what if I only consider the non negative numbers? So if I put a domain restriction and by domain restriction, all I'm doing is just throwing out certain values that I don't want to consider in the domain and in this case, those air the negative numbers. So I'm going to restrict my function F to the set 02 Infinity and I'll include zero. And so on this restriction, the function X squared is one the one. Thus it's in vertical, but it's on Lian vertebral when I consider this function X squared on this domain restriction the ray from zero to infinity. And of course, in this case, the inverse is exactly what we expected to be just square root of X. So this brings up a very subtle point that we'll see again. We have a function that is not one toe one, so it's not convertible on its natural domain. However, if we restrict the domain to a smaller set where the function is 1 to 1 than it is in veritable, and we confined the numbers. We're going to see this again when we talk about the trig and metric functions in general, Trig and metric functions will not be one toe one they're not gonna have in verses. So we're gonna have to narrow in on a small interval for which they are 1 to 1, and then we'll be able to define it embers, So it's a little tricky. But again, if you can understand this example of x squared and squared of X and realize that their Onley in verses, when I restrict the domain, then you're gonna be taking a lot of huge steps towards understanding kind of the basic ideas behind functions that we need to understand moving forward.