Common Core Algebra II.Unit 2.Lesson 5.One to One Functions

On the other hand, G of X, its rule is square the inputs. So negative two times negative two will go to positive four. Zero, we'll go to zero. And two, we'll also go to four. We don't put the four in there again. We don't list elements more than once. Now, let her be asks admittedly, a little bit of an ambiguous question. What is fundamentally different between these two functions in terms of how the elements of this domain get mapped to the elements of the range? Well, there's obviously a lot of big differences. You can say, well, the function G only has non-negative outputs, whereas function a has some things that are negative. There's lots of different things you could say. But what we're trying to get at here is that each. Domain element. For F of X gets a unique. Output. In other words. In other words, no outputs have been repeated. Now, there's nothing wrong. For function G to have an output repeated, right? The definition of a function isn't that every input gets its own output, the definition of its function, a function is every input gets exactly one output. That output could certainly get repeated time and time again. But there are some functions whose outputs never repeat. And of course, some functions whose outputs do repeat. And that's going to be what we're going to look at today. 

So I'm going to clear out this text write down anything you need to. Okay, here we go. Let's take a look at the definition of a one to one function. So one to one function. Now, the technical mathematical definition, I really like. It says a one to one function. This is one where if X one doesn't equal X two. Always means that F of X one doesn't equal F of X two. Now think about what that really means. This is saying different X's. And this is different Y's. Now you might say, what do you mean different wise? Well, remember that it's always Y equals F of X so those are Y values. They're outputs. So one to one functions boiled down to something very simple. Different X values always give different Y values always. And again, there's lots of functions that aren't like that. For instance, take our function. Sorry, it wasn't F, it was G of X, G of X equals X squared from our last example. That's a great example of a function that is not not one to one. Because different X's don't always have different wise. All right? So we're going to explore functions today that are one to one and functions that are not one to one. And you'll just kind of get used to it. All right, new category of function. Anyway, I'm going to clear the text out. And let's jump into some problems involving one to one functions. Exercise two. Of the four tables below. One represents a relationship where Y is a one to one function. Why is a one to one function of X? Determine which it is, and explain why the others are not. Pause the video now and think about this a bit. 

All right, let's talk about it. Well, remember, the deal was in one to one function is different X's get different Y's. Well, here, when we look at these, I look at the four Y's and well, they're all different. So it seems like it could be a one to one function. Ah, but this thing isn't even a function, right? Not even a function, so it can't be one to one. A function, every input, every X value can only have one Y value. But in this case, four goes to two and four goes to negative two. Same thing with 9 9 goes to three 9 goes to negative three. Let's take a look at choice two. Well, first, it's definitely a function. So all of these things, all these X values are different. But it's not a one to one function. Because both negative two goes to one and zero goes to one. So repeated Y's. No good. Let's take a look at this one. One, two, three, four. Two, four, two, four, 8, 16. Yeah, that's good. That's good, right? Each one of these X's got its own Y and really in a certain sense, each one of these Y's got its own X's. This is our winner. Let's make sure that we understand number four. Number four, problematic, because we have repeated wise. It's also problematic because we have repeated X's. All right? So one to one functions really have to fulfill two requirements. Every X has to get a single Y and every Y can only have a single X that goes along with it. All right. That's why they're called one to one. They're called one to one because if we look at a domain, then every dot over here goes with a dot over here, right? They just pair up. 

So these elements pair up one to one. Kind of like that. All right. Pause the video now. I'm going to write down what you need to. Clear no doubt. Let's keep working with one to one functions. Exercise three consider the following four graphs which show a relationship between the variables X and Y letter a circle the two graphs above their functions, explain how you know their functions. All right, well, pause the video now. There's two graphs up there of functions, and then figure out which ones which ones are functions. All right. Well, that would be choice one. And choice four. So they both pass the vertical line test. The vertical line test, right? The vertical line test that says if we slice a graph with any vertical line, it only hits at once, so no, right? And nope. Definitely not. No letter B, one of the two graphs circled. Is a one to one function. And one of them is not. How can you explain from the graph? Think about this for a moment. One of those two functions is one to one, either one or four, which one is it. All right. I'm going to go to a different color. Just so that we do it. Here it is. This is my one to one friend. All right? And it's a one to one front because you can tell that no Y values ever repeat. 

As opposed to let's say number one, there's a situation where two Y values are the same. And yet they came from two different X values. All right? So what we're looking for is no repeated Y values. All right. And this leads us to something very important. It's what's known as the horizontal line test. And we're going to take a look at that in just a moment. All right? Pause the video and copy down anything you need to. Okay, here we go. The horizontal line test. If any given horizontal line, passes through the graph of a function at most one time, then that function is one to one. All right, this test works because horizontal lines represent constant Y values. Hence, if a horizontal line intersects a graph more than once, an output has been repeated. So it's really nice because if we've got some function like this, definitely a function passes the vertical line test. We can very quickly say, oh, it's one to one. On the other hand, if we've got some function that goes like this, definitely a function, passes that vertical line test. But now, if we, let's say, draw with a different colored line. This horizontal line, well, we see those Y values got repeated. Okay? So we've got the vertical line test, which tells you whether the graph is the graph of a function, and the horizontal line test, which tells you whether it's a one to one function. But they do have to be combined together. Let's take a look at the next exercise. 

Exercise four, which of the following represents the graph of a one to one function. Think about that a little bit. All right. Well, it's this one. Choice three. Now it's important to note something. Choice one also passes the horizontal line test if I draw horizontal lines on this, they only hit the graph once. But this is not a function. All right, we can not be a one to one function if we're not a function at all, right? It fails. Fails the vertical line test. All the rest of these things fail the horizontal line test. Whoops, repeated wise. Repeated Y's. Only choice three is both a function and a one to one function. That's a bottom lines. Anyway, I'm going to clear this out, okay? All right, let's keep going. Exercise 5, last exercise. The distance that a number X lies from the number 5 on a one dimensional number line is given by this function. D of X equals the absolute value of X -5. Showed by example that D of X is not a one to one function show by example that it's not a one to one function. Now a one to one function means different X's. Give different Y's. Okay.

A non-one to one function, which will mean different X's. Give the same Y all right. So what I'd like you to do right now, then you could use your calculator to help you out too. I'd like you to try to find two different X's that give the same Y for this function. Pause the video right now and play around with it. All right. Well, there's lots and lots and lots of great choices. But here's two that work. Let's take a look at D of two. If I just substitute that into my function, right? Absolute value of negative three gives me three. Now let's see if I can figure out another one that gives me three. Well, let's try D of 8. That's going to be absolute value of 8 -5, which is the absolute value of three, which is three. All right, and that proves it's not one to one. Okay? Do you understand why? In a one to one function since these X values are different, we should have gotten different wise, but we got the same Y. And therefore not one to one. 

All right. Well, pause the video now and copy down everything you need to, and then we'll wrap up. Okay, here we go. So today what we did is we looked at a very important class of functions called one to one. Now you probably don't have a good sense at all for why they're important. We'll start seeing that a little bit in the next lesson. But functions that match an X with its unique Y have a very important place in mathematics, all sorts of different mathematics. And again, we'll be playing around with them throughout the year. Thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. Thank you solving problems.