 Hello everyone, today we're going to discuss the topic of inverse functions and this video will just introduce some of the basics that go along with inverse functions. So the first thing we need to talk about is what a one-to-one function is. So by definition a one-to-one function is a function in which each element in the range corresponds to only one element in the domain. And hopefully that definition seems a little bit familiar to you. If you think back to the beginning of the semester we talked about the definition of a function and for something to be a function each element in the domain had to correspond to only one element in the range. So what that meant is every x value could only have one y value as we went down. And so in order for something to be a function it had to pass what we called the vertical line test. So if we draw a vertical line through this graph and it only hits the graph once that makes it a function. Now for something to be one-to-one we're now adding an extra restriction. This time it still has to pass the vertical line test but now what also has to happen is each element in the range so that's like a y value can correspond to only one element in the domain. So that means we can only have each y value happen once. Here as we go across and draw a horizontal line we hit the graph twice. So this would be an example of a function that is not one-to-one. So in order for something to be one-to-one it has to pass not only the vertical line test but also the horizontal line test. And we're going to use that in order to determine if something is a function or not. And we'll use it in order to just determine whether or not something has an inverse. So the only way that something can have an inverse is if it is one-to-one. So if a function is one-to-one then the inverse of that function is a function that completely undoes the function F. It's just the complete opposite of the original function. So all of the x and y values will just flip flop and all of the operations will be the opposite and they'll be done in the opposite order. So we're going to look at a few examples of functions and then we'll look at how we could find their inverses. So if we look at this example the first thing we want to determine is whether or not it is a function. And if you look at this when we determine if something is a function we just first have to look at our input values and we just want to make sure none of them are repeated and in this case none of them are because we want each input to have just one output. So this would be an example of a function and so the domain of this function would be all of our input values. And I'm just going to list it like a set so I'll just list all of those terms in order. Not even a numerical order. I just wrote them in the order that they are there. Now we also can find the range of this function and recall the range is just going to be the output values of this function. And in this case that will be the y values here. So again I'm not listing them in order. I'm just listing them in the order that they're occurring. And if one were to repeat I wouldn't have to repeat it in the range. But there's my domain and range. Now because this is a function our next question that we have to ask is is this a one to one function? And in order to determine that we think about our range here. Were any of those values repeated? And no, we never had a repeat value. So since none of those values were repeated this will be a one to one function. And since it's one to one that means we can find the inverse. So in this case the inverse if you think back to the definition that we talked about on the previous page is the complete opposite. So here our inverse is going to just reverse all of these points. So it will contain the point four negative seven. We're just flipping the x and the y values. We're not changing the signs at all. We'll also include the point nine negative one five zero one negative two and negative five five. Okay, so that will be our inverse function of this original set of points. Now what I want you to do is pause the video and I want you to now list the new domain and range of this new function. So this function that I've listed down here. So press pause and then once you've completed it come on back and we'll look at the next example. Now here we have another example of a set of points that we're just using as our function. So we're going to do the same thing we did before. First determine whether or not it's a function. So determining if it's a function we look at our inputs. None of the input values have repeated so it will be a function. And just to repeat from before the domain is all of these x values. And notice I didn't have to repeat any while the range is the y values. So what I just look for as I'm writing out the range is if any values repeat. And what I'm noticing is I would get another seven here which I don't want to list twice. But since I got the same y value twice that is usually an indicator that this is not a one to one function. In this case we just want to look and see if these x values are the same. So notice we plug in a negative two and get seven and we plug in a two and get seven. So this value in the range the seven has two different x values. So when we answer whether or not this is a one to one function our answer will be no which means we cannot find an inverse. Now hopefully as you've worked through these few introductory problems you've been able to see a couple rules. So as we look here for just a few patterns about inverses a few things we can fill in. If f of x contains the point a b this notation here stands for the inverse of f of x. Okay so just something for you to get used to. But if it contains the point a b then the inverse will contain the point b a. Okay then a second thing to look at was with the domain and range. And hopefully you can kind of check your work for that first example here. The domain of f of x should become the range of f inverse. And the range of f of x becomes the domain. So the two will just flip-flop with each other. And those are just some key properties to remember. Usually you can figure them out once you're able to find out the inverse. But it just helps to come up with a few of those properties just from the start.