 Welcome back to our lecture series math 1050 college algebra for students at Southern Utah University as usual. I'm your professor today, Dr. Angel Misalign. This is gonna be the first video in a two two-part lecture, right? This is the first video for lecture seven. But lecture seven is gonna be talking about inverse functions, which will also continue on into lecture eight. And in this very short first video, I want to introduce the idea of a one-to-one function. So what does one mean by a one-to-one function? So we say a function is one-to-one. If whenever the y-coordinates are the same, it must have been that the x-coordinates were the same. And so the idea behind that is the following. When we first defined a function, we have something like its domain and then we have over here its range. And what we said is that for a function to be, for a relationship to be a function, we have things in the domain, in which case maybe have things in the range over here, we said that everything in the domain has to be connected to something in the range, right? You might get something like this. As long as every, there's an arrow coming out of every object, that makes it a function. And graphically, that would mean that the graph has to pass the vertical line test, that there is no point. There's no, there's no multiplicity of points which have the same x-coordinate. You can't repeat the same x-coordinate. Now, for a function to be one-to-one, we are going to require that each y-coordinate has only one arrow pointing to it. So something like this, and this is why we call it one-to-one, is that everything in the domain corresponds to one thing in the range, and everything in the range corresponds to something in, one thing in the domain. There's this one-to-one correspondence between the elements of the domain and elements of the range here. And so this, to check to see whether a function is one-to-one or not, we're looking to check using the horizontal line test, that if you have more than one, you can't have the same y-coordinate as different x-coordinates. So for example, if you take the function f of x equals x squared, if you look at horizontal lines like this one right here, we see there are violations of the horizontal line test. There's two different x-coordinates would have the same y-coordinate, and this is a phenomenon because if you take something like two squared, that's the same thing as four, but that's also the same thing as negative two squared. The squaring function violates the horizontal line test, and so this tells us that it's not a one-to-one function. On the other hand, if you take the function g of x equals x cubed, no matter which horizontal line you draw, it'll never intersect the graph in more than one location. And therefore, this graph passes the horizontal line test and is a one-to-one function. But I do have to warn you about the following situation, right? So if you were to just do one extra example here, if you had a graph that looks something like the following, you'll notice very quickly that this graph does in fact pass the horizontal line test. If we just do a couple of examples, right, this graph nowhere will intersect a horizontal line more than once. So you might be tempted to say that this is a one-to-one function because it passes the horizontal line test. Let me mention to you that this graph fails the vertical line test, so it's not a function. In order to be a one-to-one function, you must first of all be a function. It'd be all like if my wife asked me to, oh, you know, we should have a dog, right? Go find me a cute little brown puppy and bring it home as our new pet. And I'm like, okay, dear, I'll do that. I go out and come home and it's like, look, honey, I found what you were looking for. I brought home a brown snake and it jumps out of the box, right? You know, sure, you got the adjective right, but if you get the noun wrong, it still counts as wrong. In fact, she probably would be happy with any color of puppy as long as it's a puppy. You have to make sure that to be a one-to-one function, you're passing both the vertical line test and the horizontal line test.