if you remember from our first video we started out by talking about relationships then we upgraded those to functions and today we're going to upgrade our functions to better functions so how does that work let's take a look at a function and a relationship that we talked about the last time here if you'll remember our function is in orange and our relationship is in blue and if you'll remember we can write our functions with the f of x or the of x notation all right so now when we talked about functions we said that by definition every x has only one y now that's an important distinction because we said that we didn't say anything about the y's all we said was that every function every x has only one y that means that we could call this a function one comma two and we'll put another point in here two comma two and three comma two now notice that this is actually a function every x only has one y one only looks at two two only looks at two and three only looks at two so it is by definition a function as a matter of fact if we plot it out we see that it passes the vertical line test because every vertical line that goes through these three brown points only touches one brown point it doesn't touch more than one so it is a function so we can have lots of x's look at the same watt but let's talk about upgrading this today what if every y only had one x well let's see when we look at our function here our orange red function we see that every y only has up one x five only looks at eight it doesn't look at anything else and four only looks at six and three only looks at the x of four and two only looks at the x of two and what we have here is actually a special kind of function what we have is a one to one function and what one to one functions say is that not only does every x have only one y but every y has only one x and this will actually become important to us as we start to treat functions less like equations and more like operators like addition and subtraction and we'll get into that as we move further into our discussion about functions needless to say yeah we start out at the lowest level with a relationship and if you put some restrictions on your relationship you can become a function and this is every x has only one y and if you put in even more restrictions what we end up with is a one to one function and sometimes you'll see it written out one to one and sometimes you'll see numbers with the dashed and this is every y has only one x alright now of course you can't have a one-to-one function without being a function first all right you can't skip steps here all right so as we look at the hierarchy of relationships let's be honest you just need an x and a y to be a relationship but after that it becomes more special as we put more restrictions on it we see that we get into these other types of functions of which the hardest function to become is one-to-one it has the most restrictions but it also is one of the most useful functions for us as well these can be undone and we'll talk about that when we talk about inverse functions so let's take a look at a couple of functions here or a couple of equations we've got y is equal to the quantity x minus two squared plus two and that's our parabola here you can see that makes that dip here we also have x squared plus y squared is equal to 16. and that's our circle here all right and they're color coded red and blue all right as we look at these when we take a look at our first equation we can tell right away that it isn't even a function i mean you don't have to draw many vertical lines to see that oh we actually touch in two places we touch right here and right here congratulations you just failed the function test this is never going to be anything more than a relationship it's not that it's a bad relationship it's just it's just going to be a relationship when we look at our red parabola y is equal to the quantity x minus 2 squared plus 2 it doesn't take a whole lot of imagination to see that while yes these two legs are going to get very vertical they're never going to become perfectly vertical and as a result we're always going to only get a single line to pass through our function it is a function all right we've been able to upgrade our relationship the question is is it going to be ever a one-to-one function all right which would require us to have every y have only one x now if you think about the language and the test we created we created a vertical line test for every x has only one y if we switch those how are we going to switch our line in this case we're going to take it from a vertical line to a horizontal line and as we look at our function here when we draw a horizontal line through it you can see that the y of this value has an x right here and the y of this value has an x right here so there are actually two x values for this one y so it's just going to be a function it's not going to be one to one right it didn't make the jump to that one to one or upgraded function now if you've got a picture that's great but how do you tell what is a function and what isn't all right so let's take a look at our equations we already know what they are but let's take a look and see what they become so y is equal to x minus two squared plus two all right if we plug in a value for y five is going to be equal to x minus 2 squared plus 2. all right well we come across here and we subtract over the 2 and we would get 3 is equal to x minus 2 squared and to get the x by itself we would take the square root of both sides now here is where a lot of students and even occasionally mr bowen gets caught because when we're taking the square root we have to think that there are actually two answers when we take the square root we have to remember to add the plus and the minus all right and the minute we do that what we've done is we've created two answers right away as a result we know that although we can plug in any x and only get one y when we plug in a y we're going to get more than one x all right let's take the same idea and look at x squared plus y squared is equal to 16. and let's see if we can tell that it is only a relationship all right well first of all i don't know about you but i always like to have y by itself so i'm going to subtract the x squared over to the other side and to get y by itself oh there's that deadly square root again i took the square root of both sides so i must remember to take the plus or the minus of the other side all right the minute you take a square root you're going to see that that stops being a function or a one-to-one function at that point that's with even power roots remember with odd-powered roots like the cube root that's okay you don't have to add the plus or minus so as you go through and you're working with equations make sure to watch out for those squares they frequently derail plans to become a function or a one-to-one function