 We learned in the previous video what a one-to-one function is. Basically, a one-to-one function is a function which passes the horizontal line test. What that means is there will be no repeated Y-coordinates for the function that every Y-coordinate will uniquely determine its X-coordinate and every X-coordinate in the domain gives us unique Y-coordinate as well. So if this one-to-one correspondence between the X-coordinate and the Y-coordinates, in the context of a one-to-one function we can define an inverse function. If our function is called F, the inverse function is called F inverse and this is denoted as F superscript negative one. Now you have to be careful. In mathematics, we often use superscripts to talk about exponents. So like when you see X squared, that means X times X. In this context, we're using a superscript of negative one but this does not mean the negative one power. F to the negative one is not here gonna mean one over F or anything like that. It's not an exponent, it is a superscript though. And this might seem a little unfortunate. This is what one typically refers to as a homograph. That is, we are using the same written notation to mean two different things. It's kind of like if I write the word W-I-N-D on the screen right now. What's this word? Is it wind? Is it wind? It turns out it could be either one dependent on the context. And so the context has to tell you that this right here isn't an exponent because we're not taking numbers, we're taking functions. And as such, we're gonna use this F to the negative one here. F superscript negative one to represent the inverse function. Now what is the inverse function? The inverse function is gonna be the function that turns the relationship around. So we have the domain of the function over here. We have the range of the function. And we have all these things over here like one, two, three and maybe like X, Y, we'll use numbers. So maybe like seven, nine and 12. And our function F might be sending the number one to seven, the number two to nine and number three to 12. Well, what the inverse function is gonna do, it's gonna change the direction of each of these arrows. It changes the direction. So instead of the arrows going to the right, we just switch them around and they go the other way around like this now. So F inverse will send seven back to one. F inverse will send nine back to two and F inverse send 12 back to three. And so the F inverse, you know, whenever F sends X to Y, F inverse will do the opposite. F inverse will send Y back to F. It reverses the order. So if you have like an order pair X comma Y on the graph it'll reverse that order as Y comma X. Let's give a more specific example here. Let's say a function is given by the following table, X has its as, so given the function F here, its domain is 10, 14, 18, 22, 26, 30. And so that 10 maps to negative 12, 14 maps to negative six, 18 maps to negative two, 22 maps to one, 26 maps to three, 30 maps to eight, like so. Well, F inverse is just gonna switch the order of this operations. It's basically like you just switch who's on top, who's on bottom here. F inverse will then take negative 12 and map it to 10. It'll take negative six and assign it to 14. And it does that because F, it's going the opposite direction, right? F assigned to 14, the number six. F inverse will assign to negative six, the number 14. Because F assigns to 26, the number three. F inverse will assign to three, the number 26. It reverses the order of all of the assignments. So if F of 30 is equal to eight, this means that F inverse of eight is gonna equal 30. We reversed the process. Instead of sending 30 to eight, we'll send eight to 30. We just send it back to where it came from. And that's what an inverse function does. It just reverses the role of input and output. The input of the function F becomes the output of the inverse function F inverse. And the output of the function F becomes the input of the inverse function F inverse. Let's see. Let's take a look a little bit more detail about this. Let's say that we have a one-to-one function. And the reason why one-to-one is necessary is that if you reverse the direction, if you don't have the one-to-one property, the inverse relationship might not be a function and therefore not an inverse function. We'll talk about some of more of that in another video here. But let's say we know for a fact that F of two gives us thought four. That's the only thing we know about the function. We know that F will assign to the number two, the number four. Well, what is F of, what's F inverse of four? We know that's gonna equal two. Because if F sends two to four, then F inverse will send four back to two. It just reverses the direction. And conversely, right? If F of negative, well, same thing I guess. If F of negative one is seven, that means F inverse of seven will equal negative one. It reverses the direction. The X and Ys, so like the X is the number we put in, the Y is the number we get out, F inverse will just switch the rules of X and Y. The point negative one seven is on F. Conversely, negative, sorry, seven negative one is a point on F inverse. It'll switch the two numbers around. And say that we know that F inverse, what if F inverse assigns to the number three, negative five? Well, if F inverse assigns to three, the number negative five, then F will do the opposite. It'll take the number negative five and assign to it the number three. And so F of negative five is three. And so the inverse relationship is just doing things backwards. It's the relationship between addition versus subtraction, right? If you add three, then subtract three, right? One goes forward, one goes back. And we see inverse relationships all the time in mathematics, like addition, subtraction, you have multiplication and division, right? These are inverse operations. And we're just talking about, we're just trying to talk about with inverse function, the generalization of this inverse operation business.