Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. The composite function F of G the composition of F&G is defined as F of G of X or F of G of X2 different notations. We can visualize the composition of functions as follows. We're going to have an input X that's going to go into the G equation. Out of the G equation, we're going to get G of X. Now that G of X is going to go into the F equation. Then out of the F equation we're going to get F of G of X. One to one function. A function is 1 to one if different inputs have different outputs. That is, if for A&B in the domain of F with a not equal to B, we have F of a not equaling F of B, then the function F is 1 to one. If a function is 1 to one, then it's inverse correspondence is also a function. We had the vertical line test to decide if something's a function. We're now going to have the horizontal line test to see if we have an inverse or if one to one exists. So if it is impossible to draw a horizontal line that intersects a functions graph more than once, then the function is 1 to one. For everyone to one function, an inverse function exists. To find the formula for an inverse, the inverse notation is that little -1 right there F of -1. We first make sure that F is 1 to one. We replace F of X with Y. We interchange X&Y. This gives the inverse function. And then we solve for Y. We use correct notation by replacing the Y with F inverse of X. So in these problems we're going to look at F of G of one and then G of F of 1F of G of XG of F of X. So there are four different things we're doing here. When we look at F of G of one, what that means is we're going to put one into our G equation. When I put one into the G equation, I'm going to get 1 ^2 - 5 putting the one into the G equation. So 1 ^2 - 5 one squared is 1 - 5 is -4. Now we're going to put that number -4 into the F equation. So 2 * -4 + 1 negative 8 + 1 or -7. So F of G of one is -7. Let's look at G of F of one. This time we're going to put one into the F equation. So we're going to have G of putting 1 into the F 2 * 1 + 1. Two times 1 + 1 is going to give us three. Now we're going to put 3 into the G equation, 3 ^2 -, 5, nine -5, or 4. Next we're going to do F of G of X. So this time, instead of a number, we're actually going to put in X into the G equation. So if I put an XI get X ^2 -, 5 South X ^2 -, 5 is what I get out of the G equation when I put plain old X in. Now we're going to go to the F equation and put that whole parenthesis that X ^2 -, 5 in every time we see our unknown. So we're going to have two. Instead of sticking in a number, we're going to put an X ^2 -, 5 + 1. So we're going to get two X ^2 -, 10 + 1, and that's going to give us a final of two X ^2 -, 9. If we look at the last one, that was G of F of X. So we're going to put plain old X into the F equation, and if we put X into the FF equation, we're going to get 2X plus one. Now we're going to put the two X + 1 into the G equation, so two X + 1 ^2 -, 5. So if we have 2X plus one squared, we're going to foil that out and we're going to get 4 X ^2 plus four X + 1 -, 5. If we combine our like terms, we're going to get four X ^2 plus four X -, 4 SO4 parts to that first problem. Let's look at #2 we're going to do all four parts again. So we're going to start with figuring out what F of G of one is. If I want F of G of one, I'm going to put one into the G equation. So we're going to have sqrt 1. Well, sqrt 1 is 1. So now we're going to stick that one into the F equation and get 10 -, 1 or 9. That's the first part. The next part says find G of F of one. So now we're going to stick 1 into the F equation, so 10 -, 1, that's going to give us 9. Next we're going to stick 9 into the G equation, so sqrt 9, which is 3. The next part set F of G of X. So we're going to put X into the G equation and get out square root of X. Now we're going to put square root of X into the F equation. So every time we see our unknown, we're putting in the square root of X. So 10 -, sqrt X is our final answer. For the next one, we're going to have G of F of X and that's going to say G 10 -, X is our F of X equation. So now in the G equation, every time we see our unknown, we're going to put in that 10 -, X. I'm going to let you try #3 on your own. These next type of problems, we want to look for compositions. We want something inside of something. So we want F of G of X to equal 3 X -1 quantity squared. So if we wanted to think about this G of X function inside, we look over here and we think about what's inside, and the inside portion is that three X -, 1. So if we have three X -, 1 as the inside portion, then we're going to think about that inside portion all being replaced as an X. If that whole thing is being replaced as an X, we end up with just X ^2. So our G of X is going to be 3X -1 and our F of X is going to be X ^2. So once again, we're going to look for a composition in #5. I want F of G of X to equal sqrt 5 X +2. So what's inside? Well, inside G of X or the inside of this equation would be the five X + 2. If I take that whole five X + 2 out, that leaves my F of X equation to be square root of whatever I took out, which we're going to signify as plain old X. This next one is a little tricky and there are more than one answer for all of these. I'm just showing you the most obvious one. So here we could think of a fraction plus something. So if we let G of X equal the 3 / X, then the F of X, when I take out that 3 / X and replace it with my unknown would end up being X + 4. These next ones we want to first of all determine if it's one to one, and if it is one to one, we want to find the inverse. Well #7 is a line, so it's got to be 1 to 1. So we're going to switch the X and the Y and then we're going to solve for Y. So we're going to add 3 and divide by two. So we're going to get X / 2 + 3 / 2 equaling Y. Using our correct notation, we have F inverse of X equaling X / 2 + 3 / 2. This next one is also a graph of a line. So we're going to switch our X and our Y and solve. So we're going to start by subtracting 2, and then we're going to multiply by three. So three X -, 6 equal Y using our correct notation F inverse of X equal three X -, 6. The next one is a parabola. If we think about this as a parabola, it wouldn't pass the horizontal line test because there'd be at least one line I could draw in that would intersect the graph more than once. So not one to one. Looking at the next one, this one is a horizontal line and hence a horizontal line can't be 1:00 to 1:00 because it doesn't pass the horizontal line test. So not one to one for the next problems, we're going to graph the function and its inverse. So we have X ^3 - 1. If we have X ^3 - 1, we could make ourselves AT chart put in a couple positives zero and a couple negatives. So if I put in two and we go 2 ^3 - 1, two cubed is 8 - 1 is 7 1 ^3 - 1 would be 0 0 ^3 - 1 is -1 negative 1 ^3 - 1 -1 ^3 which would be -1 - 1 would be -2 -2 ^3 - 1 would be -9. Now the inverse is just really switching X&Y. So if we switch our X&Y, our inverse function would be the .7201 negative 10 negative two -1 and -9 negative 2. So if I'm going to graph 721234567201 -1 zero -2 negative one and -9 -2, if we graph this, it's going to look something like that. If we graph the other one, we have the .27. We have the .100 negative one, negative one -2 and -2 negative 9. If I connect those points, something that looks like that now because they're inverses of each other, if I drew in the line Y equal X and folded this graph along Y equal X, the two parts of the graph would fall right on top of each other. For the last example, we're going to use square root of X. Well, if we're using square root of X, we can only have positive numbers and 0. So if we have 0149 square root of 0 is 0 sqrt 1 is 1 sqrt 4 is 2 sqrt 9 is 3 The inverse we're going to have the X and the YS flipped. So for the inverse, we're going to get 00 and 1124 and three nine, so 001124 39. If I connect those, we're going to get a graph that looks something like this. If we graph the original, we have 00, we have one, one, we have 4 two and 9:00 three. It's going to look something like that. If we put in our line Y equal X and folded this graph along y = X, the two parts would fall right on top of each other. Thank you and have a wonderful day. This is Anna Cox.