 In the previous video, we just went over a short introduction of one-to-one functions and then also inverse functions and this was the definition, just the informal definition of an inverse function that I gave you. So recall that an inverse function is the complete opposite of the original function. So that means it includes the opposite operations of a function but also the key thing to remember is they also have to be done in the opposite order. So in this video, we are going to look at finding inverses to functions when we're given the equation. First, we'll just look at how to do that using this idea of the opposite operation in the opposite order, but then we'll also look at a method to do so algebraically. If you look at this function, what we want to think about are what operations are being done to x right here? So if you were going to plug in a value for x, what would you do first? And the first thing that we would do is we would multiply that value by 2 and then we would subtract 3. So what we have to remember when we're finding the inverse of this is we're going to do the opposite operation in the opposite order. So we start from the bottom and that becomes our first operation. So we're going to start with the opposite of subtracting 3 and the opposite of subtraction is addition. So we first will add 3 and then we're moving our way up this column here. And so the next thing that we had done was multiply by 2 and the opposite of multiplication is division. So we will then divide by 2. So how do we write this out as an equation? Well, we start with our x value. We add 3. That was our first step and then we're going to divide all of it by 2. So you just have to make sure that you are applying that second operation to the entire function. Okay, let's look at a second example here. Let's look at g of x equals the square root of x minus 4. We're going to start by listing the operations in order that are being done to x. So if you look here's our x and if we were evaluating this function, the first thing we would do to evaluate is we would first subtract 4 and then the second thing we would do is we would apply the square root or take the square root. Now recall the inverse is the opposite operations in the opposite order. So we start at the bottom and we work our way up. So the opposite of taking the square root is going to be to square the function and then as we work up the opposite of subtracting 4 will just be to add 4. And so our inverse will be x squared. That's our first thing and then we add 4. Now both of these examples are fairly basic in finding the inverse. There are a few examples that can get a little bit trickier. So we are going to look at a couple examples. Again, they're too hard, but we just want to look at how we could find the inverse algebraically. So what I first want you to do is on your paper, we first want to find the domain and the range of this function. So if you want to, you can type it into your graph. Just pause the video, type it in just so you can look at it. But remember the domain is all of our possible x values. And if I just sketch a little picture of the graph here, it should look something like that. The domain of this function is going to be all real numbers or an interval notation, negative infinity to infinity. While the range of this function is from the bottom to the top and these, although they are increasing and decreasing just slightly, these will continue to increase or decrease. So our range is actually the same as the domain. So if you look, this graph passes both our horizontal and our vertical line test. So it is one to one, which means we can find an inverse. So our next step is going to be to find the inverse. Okay, now here's a way that you can find the inverse just algebraically if you don't like using the idea of the opposite operation in the opposite order. Instead of writing this as f of x equals, I'm going to just write it as y equals. And if you remember from the previous video, when we're finding the inverse, the x and the y values just flip-flop. So in the equation, we can just trade the x and the y values and now we just need to solve this new equation for y. If we can do that, then we'll have our inverse equation. So in order to solve for y, start from the outside and work your way in. Here, to get rid of a cube root, I would cube each side of the equation. That would give me x cubed on the left and then the cubed and the cube root cancel out. So I would get x cubed equals y plus 2. And then to solve the next step for y, I would subtract 2 and I end up with x cubed minus 2 equals y. Once the y is by itself, you have come up with your inverse equation. f inverse, I'm trying to use our notation, equals x cubed minus 2. So our second example is g of x equals x squared minus 3. So again, I want us to start out by finding the domain and range. If you want, pause it, type it into your graphing calculator and we'll take a look at it. Here, I'm just using my ideas of transformations in order to get a rough sketch on here. So if you look at this, we have an issue that hopefully you see right away. If I were to draw a horizontal line through this graph, it hits the graph twice, which means it is not a one-to-one function. And if you remember, functions can only have inverses if they are one-to-one. So what we can do in order to still be able to find the inverse is something called restricting the domain. And what that means is you want to take just part of the graph that still gives you every single y value possible, but it makes it so that you only get each y value once. So here, if I look at it, if I start down here at the vertex and I just include this half of the graph, just that red piece of the graph will pass the horizontal line test. So I'm going to restrict the domain, and rather than saying my domain is all real numbers or negative infinity to infinity, I'm going to restrict it and start right here so that I can actually find the inverse. So my domain is going to go from zero to infinity, and then the range, it should still include all of the values on the range of the original function, otherwise you should pick a different piece of the graph to restrict it. But here our lowest value is negative three, and the highest is infinity. So when we find the inverse, even without finding the equation for the inverse, we already know what the domain and range of the inverse will be. The domain from g of x will become the range, and the range will become the domain. So the domain of our inverse will be negative three to infinity, and the range will be zero to infinity. And those you can just check at the end, but let's find the actual inverse algebraically. So we'll use the same process that we used in the previous example. We start by switching our x and our y. Next we're going to solve for y. So first in this case I would add three to each side of the equation, and then next I am going to take the square root of each side of the equation. So what I end up with here is I end up with the square root of x plus three equals y. Or in other words, g inverse equals the square root of x plus three. And if you were to just sketch a quick graph of that, or type it into your calculator, it's our square root function shifted to the left three. And if we shift to the left three, that gives us a domain of negative three to infinity, which is what we thought would happen, and a range of zero to infinity, which is what we thought would happen as well.