 This video will talk about inverses and one-to-one functions. A one-to-one function shows that every element in the range corresponds to only one element of the domain. If you remember when we talked about functions, we were saying that every input or domain value had only one range value. So one-to-one is going back the other direction. And if we look at this, we have the horizontal line test here. The very line intersects the graph of the function and at most one point the function is one-to-one. If you remember to find out that it was a function, we had the vertical line test. So we could go something like this. Now we have the horizontal line test and it says if I go this direction, how many times do I hit my graph? And if I look at this one, I can say that it's one-to-one because it passes that horizontal line test. If I come over here, this is a function. It's a quadratic function. We know that function. But if we do the horizontal line test, we find out that it is not one-to-one. And then we come back over here and this looks like a sideways quadratic and it really is. It's x equal y squared instead of y equal x squared. And if we do the horizontal line test again, we are going to get that we only hit our graph once. So this one again is not one-to-one. So inverse functions. Inverse function has to be one-to-one and it's going to be a one-to-one function that has order pairs AB. So the inverse function and this is inverse function notation right here. And I want you to understand, you've probably heard this before, but this is not f inverse of x is not equal to 1 over f of x. This doesn't mean f to the negative first. This is just the way we write inverse function notation. When we look at this, if we have a function that's one-to-one with order pairs AB, then the inverse function is going to have ordered pairs BA. And if the range of f will be the domain of the inverse function, and the opposite is also true, the domain of f will be the range of the inverse function. And that's why these A and B points switch. So it asks us to find the domain and range of this function. Well, the domain is going out forever and in both directions, left and right. So the domain is going to be all reels. And the range is also going down forever and up forever. So it is also all reels. Then it says, determine if the function is one-to-one. Well, if we do our horizontal line test, we can say that it is not one-to-one. All right? Now if we look at this function, it happens to be an ordered pair. But what's the domain? Well, the domain in our case here then is going to be x is an element of, goes from negative 6. We have negative 3, 8, 12, and 2. And the range is going to be y as an element of 2, 7, 0, negative 1, and negative 3. Probably would even be better if we had braces on here. It's just a set of numbers. And we want to know then if it's one-to-one. Well, if we look at our range values, we had five different range values and we've had five different inputs. So that means that every input had its own output, made it a function. And every output had its own input, so that made it a one-to-one function. Remember, it must be a function first to be one-to-one function. And then it says find the domain and range of the inverse function. Well, the domain here, x is going to be an element. And this time I'll write it nicely. We'll just make it a set. It's going to be these range values because it switches. So we have 2, 7, 0, negative 1, and negative 3. And our range is going to be our domain of our function. So it's negative 6, negative 3, 8, 12, and 2. Okay, it asks us to state the domain and range of this function. Well, the domain is going to be, we can put any x in here and multiply it by 3 and divide by 4. So the domain is going to be all-reals. And the range, no matter what I put in here, I'm going to get a real number out. So the range is also going to be all-reals. Find the inverse function and state is domain and range. All right, so here we go. How do we do that? There's a couple of ways you could do it. One way is to do what I call the table method. And let's just start with x and then just list order of operations. Well, the first thing we would do is we multiply by 3. And the second thing we do then, after we've multiplied x by 3, is we divide it by 4. So we want to go the opposite order and opposite operation. So the last thing we did was divide by 4. So now we're going to take x and we're going to multiply by 4. The first thing we did was multiply by 3. So the opposite of that is to divide by 3. G inverse of x is going to be starting with x and we multiply by 4. And then all of this is going to be divided by 3. And when I'm doing the table method, I like to put parentheses around what I've done. Because each step that you do is applied to everything up to that point. The other way to do it is what we call the algebraic method. An algebraic method says that we want to switch the x and y. Because remember we had in our inverse function, if this was x, y in our function, then the inverse function was y, x. So now instead of g of x, which is really y, we're going to say that's x is equal to 3y over 4. And then we're going to solve for y. So we have to multiply both sides by 4 to clear the fraction. And then we're going to divide both sides by 3. So we have 4x over 3 is equal to y. And then you just put it in inverse function notation. So I would have g inverse of x is equal to, and notice part of the reason I switched my x and y was because I would have an x in my answer and my function is of course going to have b with respect to x. So here's my inverse function. Either way I did it, I got the same thing. And then the domain, again, is going to be all reels. And the range is going to be all reels. And I remember that the domain is really the range of your inverse function and the range is really the domain. Well, it's not easy to see here because everything is all reels. But we do want to graph it. So let me call it my calculator here real quick. And we want to put these two equations in as their own equations. The first one was 3x that was being divided by 4. And the second one was going to be 4x divided by 3. And if we use a standard window, we get this graph. And if I could put it in here, I'd put in a dotted line for y equal x because they're symmetrical about y equal x. And here's how you know that. y equal x. That means when x is 4 and the function y is 3, we'll go make this y now a domain value. So when x is 3 and my other function, the inverse function, y is 4. So x equal y, y equal x. So that's symmetrical about that line y equal x.