 This video is part two of inverses. We're going to find the inverse of a function algebraically right now. So we're going to switch the place of the x and y, remembering that y is equal to f of x, because it's going to look like function notation. But just remember that f of x is really a y over there. Then we're going to solve for y after we've switched the places. And then in the end, we'll replace y with f inverse of x, because remember, function notation is the same thing as a y. So let's think about this problem. This really says y is equal to x plus 5 over 4. So if we rewrite it, now we have x is equal to y plus 5 over 4. We've switched our x and our y. So to solve, we clear the fraction first. 4x will be equal to y plus 5. And then we subtract 5 from both sides. So 4x minus 5 is going to be equal to y. And then we just rewrite y with an inverse function notation. Now for some people, that bothers them a little bit, because we're so used to seeing f inverse of x first. So if it helps you, you could rewrite this as y. That's the right-hand side. Now it became the left-hand side and the left-hand side, just like it looks, is going to become my right-hand side. And then you can say that f inverse of x is equal to 4x minus 5. Or you can just replace it. It's not wrong to say 4x minus 5 is equal to f inverse of x. We don't usually put it last. So let's try again. So here's my y, and this is x squared. So when I rewrite it, this is going to be y. But x is being squared, so y is going to be squared. And then minus 4 goes on that side. And on the other side, instead of y, we're going to put our x. So I'm solving for y, so I have to add 4 to both sides. So x plus 4 is going to be equal to that y squared. And since it's y squared, I need to take the square root so that I can cancel the square and the square root to get y. But if I take the square root on that side, then I have to take the square root on the other side. And so I take the square root of x plus 4, and I have to leave it as a quantity, because I don't know what x is. I don't know that it's necessarily going to give me a perfect square. So I just have to write it as y is equal to the square root of all x plus 4. Now we're ready to replace the y with f inverse. And what was y equal to? y was equal to the square root of x plus 4. Again, if you needed to, you could rewrite this as y. Put it from the right to the left, and on the left-hand side, now it becomes your right-hand side. y equals the square root of x plus 4. And then rewrite y with the inverse function notation. This is our function. f of x is equal to x plus 5 over 4, and the inverse function, if you remember, was 4x minus 5. So I want to graph both of those on my calculator. So in parentheses, I have to write the x plus 5, remember, because it's more than one term up there, divided by my 4. And then y2, I'm going to have 4x minus 5. And if I graph it, it says, what do you notice? And some of you may be saying, I really don't notice anything. Well, let me sketch the graph for us. So here's my graph, and it looks something like this. It's not perfect, but it looks something like this. Well, if you look real close, they kind of are mirror images of each other. At this point here, it looks like it's about like that one there, and this one here looks like it's about that one there. So they're mirror images, and it's actually a mirror image across this line that happens to be y equal x. And now they want us to look at the table for these two functions. Well, we've already got them in our calculator, so we might as well just go look at our table, and where are we starting? They want us to start, make our tables here. So when x is negative 1, y1 was my function. That's what they ask, is the original function of y1. So negative 1, y is 1. And then they want x is 3, then y is 2. And we have to arrow down a little bit and find out that when x is 7, for our function, y is 3 up here, and then 10, 11, we find out that when x is 11, y is 4. And if I do the inverse function, that's my y2. So I come back up and look at it a little closer. When x is 1 over here at y2, it's negative 1. When x is 2 over here at y2, I have 3. x is 3 over here at y2, I have 7. And x is 4 over here at y2, I have 11. And if you notice, this is the ordered pair, negative 1, 1. And in the inverse function, I have to pair 1, negative 1. This is 3, 2. The inverse function has 2, 3. 7, 3. Inverse 3, 7. My function has 11, 4 as an ordered pair and my inverse function has 4, 11. So what did we notice? The x and y change place. They switch. And if you think about the y equal x thing, the y has now become the x and the x has now become the y. Here in my inverse function came from the x of my function and the y of my function is now the x of my inverse function. The x's and y's switch places. All right. So hopefully you've noticed that from those graphs that when you have an inverse, the x's and y switch places and that means that the inputs and outputs are going to switch places. That's what we saw in our tables. This also means that the domain and range also switches since they correspond with the inputs. The domain, if it's A, B, and C, if the domain happens to be A, B, C and the range happens to be D, E, F, then the inverse function is the domain will be D, E, F and the range will be A, B, C. It also applies to applications. So imagine that C of H determines the cost in dollars of H hours at your college. So the inputs of C of H are, remember it's the cost function in H hours. So it would be this H is going to be our hours in our function. And the outputs are the amount of dollars that we have to pay. Well remember that we just talked about the fact that the domains and ranges are going to switch. So they mean the opposite things. C inverse of H is actually going to be an hours equation and H is actually going to mean dollars. So the inputs are now dollars and the outputs are hours. So let's think about the meaning then. This is our function and this is an H which is hours and this is the outcome which is dollars. So it means that five hours credit hours cost $300. And when we do the inverse function, this is no longer hours. This is now dollars and this is hours. So everything switched. So to interpret four credit hours is going to cost $200.