 So in this lecture so far, we've talked about what is an inverse function and how to recognize when two functions are inverses of each other. We're ready to start beginning the discussion of how does one construct an inverse function given the function that you started off with, right? And it really comes from the following observation. This is, again, a recognition statement right here. If you have a function which is one-to-one, like the yellow function you see illustrated on the screen right here, the function f, this function is in fact one-to-one, you can see that it passes the horizontal line test. If you have a one-to-one function, you can, if you take that function, since it's one-to-one, it'll have an inverse function, which you can see illustrated in green right here. These two functions f and f inverse will satisfy an interesting symmetry property. If you take the line y equals x, right, you see that's this dashed line that goes to the diagonal here. The line y equals x, now the significance of this line y equals x, y equals x, this is the graph of the identity function, y equals x. If you take the diagonal line, the identity line y equals x, the function and its inverses will be mere images of each other, exact reflections across this line. This is the geometric interpretation of the inverse function property we saw before. We saw earlier that f composed with f inverse of x is equal to x, the identity function, and f inverse composed with f at x is equal to x as well. The inverse function property, inverse function property here tells us this, when you interpret that geometrically, you see exactly this picture right here, they're mere images of each other. What's happening here is that the roles of x and y are reversed with one another. Like for example, if you take the x-intercept of this function right here and you reflect it to the other side, this is an x-intercept, then if you reflect it to the other side, you're going to end up with a y-intercept. So the y-intercept of f corresponds with the x-intercept of f inverse, and likewise if you take the x-intercept of f here and you reflect it across this line to the other side, you're going to get the y-intercept of the f inverse. And this is what happens all the time, if you take any point right here, you take a point x, y, then you reflect it to the other side, you're going to get the point y, x, you're going to switch the coordinates around, x becomes y, y becomes x. That's what this symmetry principle is telling us. And so we can see whether a function is an inverse of another because it's a reflective property. Now let's suppose we don't actually know what the function is, or the inverse function is. If we have a graph, let's say something like this, if we have a graph and we have a function which is say one to one, maybe it looks something like, I don't know, it'll do something like this. If this is our function, we'll just consider the diagonal line, y equals x right here. The inverse function, assuming this here is f, the inverse function would be the reflection of this graph across that diagonal line. So it would look something like the following. This would be f inverse. That's how we can construct, at least geometrically speaking, it should be a little bit closer to the diagonal. It's not perfectly drawn to scale, but we get something like that. That looks a little bit better, f inverse. So these should be exact mirror images of each other across the line y equals x. So let's look at some specific examples. Let's take a look at an example we saw earlier. In a previous video, we showed that the function f of x equals 2x plus 3 has its inverse function, 1 half x minus 3 half. So we composed these and showed algebraically their inverse functions of each other. Notice the reflective property. These two lines are mirror images of each other across the diagonal line y equals x. The y-intercept of 3 corresponds to the x-intercept of 3 for f inverse. And likewise, the x-intercept of this function, what is that, negative 1.5? That'll correspond to the y-intercept of the inverse function at, again, negative 1.5. And if you take any point along the way, so we have this point on the graph here, negative 1 comma 1, if you switch the points around, you're going to get on the inverse function, 1 comma negative 1. So the points on f, you swap the x and y coordinates, that'll give you a point on f inverse. These functions will intersect each other on the line y equals x. That's where these things will intersect one another. Here's another example. Consider in yellow right here, the function y equals the square root of x minus 2. That's illustrated here in yellow. If we reflect this graph across the diagonal, y equals x, we get the graph of the inverse function. Notice there's this vertex point, the x-intercept 2 comma 0. That's going to be the y-intercept of f inverse, 0 comma 2. This function then proceeds to increase, right? So as x goes to infinity, y is also going to go to infinity. We're seeing that same behavior all the way it gets swapped around. So you have this one, it's going up, up, up, up slowly. This one's going up and up and up more rapidly. The rate at which this function is increasing is a lot faster than this one. That's because they're mirror images of each other. One thing I do want to mention here is that with the function f here, the domain of this function is going to be 2 to infinity because there's nothing defined to the left of 2. And its range would be 0 to infinity. It's going to get only positive numbers. For f inverse on the other hand, what you see on the graph here is its domain is 0 to infinity, like so. And its range is going to be, looking at the y-axis, only these points right there, its range is going to be 2 to infinity. And look at what happened here. The range of f inverse is exactly the domain of f. And also the domain of f inverse is the range of f. And this is not a coincidence. This is what's going to happen all the time. If you take the domain of some function f, this is going to equal the range of f inverse. And likewise, the range of f equals the domain of f inverse. Because what is the domain after all? The domain is the set of all possible input of the function. It's the set of all possible x's. And the range is the set of all possible y-cord and set is the set of all possible output. If the inverse function switches the role of input and output, the output of f inverse is the input of f. So the range, which counts the stuff that comes out of f inverse, is exactly the stuff that can go inside of f and vice versa as well. So this reflective property is something we see with a function that's inverse. It'll be reflective across the line y equals x. And in fact, you can draw the graph of f inverse just by reflecting the picture onto the other side. In the next lecture, lecture eight, we'll talk about how we can use, how we can turn this geometric property into an algebraic property and give us an algorithm for computing the formula of an inverse function.