 Here's the rewrite of system talking about the plus and minus. The solution to x squared is equal to 4 is plus and minus 2. You need both solutions to be correct, so you're saying, yes. I'm just saying we take square root of x to represent only the positive, because if you don't, it's not a function. Yeah, for sure. If you're talking about functions only, 100%, right? Function. Square roots having two positive outputs would make there be four numbers the quadratic formula would give us. Four numbers, it would give you two numbers, like square root of da-da-da-da-da, would be two values, wouldn't it? Yeah, check this out. Here. Now, what what system is saying is 100% true when we're talking about functions, right? So this comes into play here. Watch this. I'm gonna kick it down for you. One of the first places we encountered this, right? Let me see if I got a pen that's gonna be a little darker. Better coming out. Is that better? Yeah, maybe. I need a whole new one. Let's find the pen that's gonna be nice and dark. That's not it. Let's dump that one. Oh, it's not bad. Brown? We'll use brown. There, we'll keep this one here. Let me see the red one too. Oh, that's nice. That's darker. Let's use the red one. Okay, now take a look at this thing. Talking about why a function would not be a function or a relation would not be a function if you had a positive and negative outcome when you took the square root of a number of square root of function, right? Now, one of the things that happens in mathematics is we try to manipulate functions, change things around to see what happens to them, right? So just imagine, just imagine you had this. y is equal to x squared, right? Let's assume we wanted to graph this function. Now, graph of this function is a quadratic. We've done a lot of quadratics before, or you could just create a table, right? So let's just create a table. Just so for those that want to follow what's going on here, you don't have to know about quadratics and the quadratic formula and completing the square and stuff like this to be able to graph this. So this is just a function, right? A function is just sort of a relationship but a special type of relationship where for given x, you can only have one y, right? For a given input, you can only have one output and right now we have a y is equal to x squared and the x we consider to be our independent variable. It's our input and y is our output, right? So if we want to graph this function to see what it looks like, all we got to do is put in an input and get an output, right? Put it in, get it out. So let's put something in for x. Let's put in zero for x and all you do you say, okay, y is equal to zero squared. What's zero squared is zero. Okay, so that's a point on the graph, zero zero. Let's put another input, one. When x is one, we get one squared, we get one. So when x is one, y is one, x is one, y is one, right? Let's put another point, x is two, right? Well, if you put two for x, you get two squared, that's four. So when x is two, two, three, four, right? Oops, let's graph this better. So we're here, right? Let's put in three. When x is three, you get three squared, you get nine. That's four, five, six, seven, eight, nine. We're up here. We're off the board. So let's just graph this, right? Here's what it looks like on this side. Well, it goes up like that, right? Well, okay, we've got a feeling for what the graph looks like on this side of the y axis, right? Because this is, this is your y axis, this is your x, right? That's where your x, x this way, y this way. Well, let's see what it'll be if x is negative one. What's y when x is negative one? So you plug in negative one for x, right? So you're going to get negative one squared, which is one. Oh, one again, so one. And if you plug in negative two, you're going to get negative two squared, you're going to get four. Oh, so that's a mirror of that, right? Cool. Hopefully that's symmetrical enough, symmetrical enough, right? So that's a graph of your base quadratic function, which is the parabola, which opens up like this. Cool enough. Now, what a mathematician do in their infinite insanity? They go, hey, what'll happen if we switch the x and a y here? Switch them up, switch them up. What do you mean switch them up? Well, make the y and x and the x or y. Y for the hell of it. Let's see what happens. Okay, let's do it. So what happens if we do this? Fine. Yes, I want to do a different color, so we know it's the different stuff. Green, brown, fine. Take, take, I don't want to write all capitals, take inverse of y. Here, take the inverse of it, which means flip the x and the y around. So what you get is x is equal to y squared. Flip the x and the y. That's what an inverse means. What it also means is a reflection about the line y equals x. So you're reflecting a line. I should make this one. Give it a little curve, right? Because there is a parabola after all. Give it a little curve. Give it a little curve. So when you take the inverse of a function, what you're really doing is you're doing this. You're taking a function and flipping it about the line y equals x. Y is equal to x. You're flipping it about that line. So, okay, let's do a little algebra on this. Well, if you're going to do a little algebra, you're going to get y by itself. Y is your function. Y is your independent variable. x is your dependent variable. Oh, sorry. Y is your dependent variable. It's dependent on x, so you want to get y by itself, right? So you take the square root of both sides. So y is equal to square root of x, right? But what we talked about was square root has to be plus and minus, right? So, square root of anything is plus and minus. Now, remember, we don't have a number in there yet, right? So do we have to write in plus and minus here? Not really, because we haven't taken the square root of x yet, right? It's just a variable, right? By definition, you take the square root, it's plus and minus, right? So let's leave it like that. Let's not put plus and minus there, okay? Now, if this is what's going on and you're taking the inverse of this function, which means you're reflecting about the line y equals x, which means all you're going to do is flip the x and the y. You're switching the numbers, right? If you're going to switch the numbers, let's graph it. Let's create a table. Well, if we're going to create a table, all we're going to do is flip these guys. So the x becomes a y, so 0, 1, 2, 3, negative 1, negative 2, and the y becomes an x. 0, 1, 4, 9, 1, 4, okay? We can test this if you want. Right? What's the square root of 0? Put 0 in here. Well, square root of 0 is 0. What's the square root of 1? Square root of 1 is 1. What's the square root of 2? Well, square root of 2 is just square root of 2, right? What's the square root of... I'm putting this in the wrong way. The square root has to go here. So over here, y is equal to... No, no, that was correct. Y is equal to square root of x. So square root of x is equal to 1. Square root of 2 is equal to square root of 2. Square root of 3 is equal to square root of 3. Square root of negative 1... Oh, no, no, I'm already putting the... Here. Square root of negative 1. Square root of 1 could be negative 1, and square root of 2, square root of 4. Look at this. Look at this. I'm messing this up. 4 is 2. Square root of 9. Did I confuse guys enough? Square root of 1 is negative 1. Square root of 4 can also be negative 2, right? I don't usually do it this way. I'm trying to push it, right? So square root of 4... You put 4 in for x. Square root of 4 is 2, right? But it's not just 2. It's plus and minus 2. So instead of putting minus here, I'm going to put it here. Plus and minus, right? Square root of 9 is not just 3. It's plus and minus 3. Square root of 1... Well, it was 1 and negative 1, so it's plus and minus 1, right? So we don't need these bottom guys. We can just take them out, right? I hope that's clear. I sort of mucked it up in the process, right? Crafter, how you doing? Hope you're doing well. So check this out. What does this mean? That means if we're going to graph this, when x is 0, y is 0. So we're here. When x is 1, y is plus and minus 1. When x is 1, y can be here or it can be here. When x is 4, 3, 4, y can be plus and minus 2. Here and here, right? And then 9 is plus and minus 3. We're off the board again, so let's graph this. Are we snagging anything today? I got some grapes. They're pretty good grapes. I got some that are loose here. They're really nice grapes. Just a little. I had a good breakfast. Let's see our grapes. Who's going to focus on my grapes? There you go. Really yummy grapes. Oh, I'm going to look like it's got a little well done. Oh no, it's just the end of it. So it's pretty yummy. As well as learning how to calculate my hand. No, I would not recommend high definition grapes. High definition grapes. So take a look at this thing. This is what the inverse means of this function. Now remember, I keep on calling the function, but you can think of it as a relation if you want. So if you take this function, this relation, and take its inverse, that means switch the x and y around, switch the x and y around, means you're taking this function and flipping it along this, you get this. Now here's where the problem comes in, a system saying the plus and minus. If we say this is a function, then its inverse for it to stay a function means that it has to pass the vertical line test, which means for a given x value, it can't have two different y's. For a given x value, x is equal to one. It can't have plus or minus one as an answer. For x is equal to four, you can't have plus and negative two. You can't have an x pointing to two different y's. That's what it means for it to be the definition of function. If the question was, this is a function, then you have to decide if you're going to kill the top or kill the bottom of this, depending on your system. Usually you kill the bottom, and you say, oh, which means you're killing all the negatives, all the negative values. Which means the inverse of this function is going to be this function, and this function looks like that. So you don't have the negative results when you take the square root of a number. However, if I said find the inverse of this relation, then if I define it as a relation, that means this can be a relation, that means the negative numbers can remain, which also means that these numbers would still be there. So it's all about definition. It's all about definition. Usually in high school mathematics, you're dealing with functions. So you end up eliminating the bottom. But let's assume you have this. Let's see if I was green pens doing. What if you had this? y is equal to negative square root of x. If y is equal to negative square root of x, then your graph would no longer be the top guy, it would be the bottom guy. Because the square root of x, you would define to be positive, and you're taking the negative of it. So that part would be the legit answer. Okay. Does that clear things up? Is that okay, system? Or anybody else that's wanted to know what this is about? It's interesting. It really digs down deep into the essence of what it is that we're doing. Most people don't appreciate this. And system, thanks for bringing it up, by the way, it's important to have a visual of what it is really that we're doing and why it is that we're doing it. How does the syntax work? And it's all about the word. Function. Function. Oops. Function. y is a function. That would mean take inverse of y, which is a function. Right? If I say this is not a function, or if you don't specify, right? I think it's legit to put it that way. It really depends on the teacher's as well and the correction. Like, that's the kicker with this, right? It's really dependent on how you're learning it. Right? But the syntax, the math, it's just there. It's just there. That's the thing with mathematics. A lot of, unfortunately, they make special rules in math to apply to a certain system, and people think those special rules are universal and you can do that in the language of mathematics whenever you come across it, but that's not true because that applies only to that system, right? So, for example, when you're doing calculations, if you're finding maximum area, maximum area in general, right? Your graphs for maximum area when it's quadratic, you go from here to here, right? You end it there. Your domain and range, right? Why? Because you're still solving for quadratics, but you can't have a negative area when you go down this way, right? You can't have a negative area, so you eliminate anything below the x-axis, okay? Because you can't have a negative value for that, right? That's the definition of the system. However, for quadratics, you can have negative numbers, right? That's their infinite, okay? I hope that helps out. I love talking about this stuff because it gets into the nitty-gritty of what it is that's going on.