Hello wonderful inquisitive math people. We're going to look at functions with algebraic operations. If X is in the domain of two functions F&G, then we have several things that are true. We can add the two functions together. If we put parentheses around each of those functions, it really means F of X + g of X. We could subtract the two functions. Once again, the parentheses tell us that those two functions are together and X is our input or our domain of F of X -, g of X. We could have multiplication of the two functions F * g of X is really just F of X * g of X where X is the domain. Or we could also have F / g of X where F of X is divided by G of X, where X is the domain and where G of X can't be zero because we can't divide by zero. We're going to look at some different kinds of problems. One is if you're given a graph and we want to figure out from the graph some different things that may be true. The first thing I want to look at is the domain of F of X, the domain of F of X. The domain of F of X means we need to come over here and we need to figure out how far left and right our X goes. So we've gone 12345678 to the left, so -8 less than or equal to X. And then we need to know how far right we've gone 1234. So we know that our X value is in between -8 and four. If we look at our G of X, we're going to talk about how far left and right it goes. So 123 to the left, 123-4567 to the right. So the domain of G of X is -3 to seven. That means my X value has got to be somewhere between or on the end points of -3 for the X to seven. So -3 to the left side, 7 to the right. We can also, and this one's the important one, look at the domain of if we combine these functions. So if we took F + g of X or F -, g of X, the XS had to be defined in both functions. So when we look here, if we came out to say -5 are there both an X for the F function and the G function out here? And the answer would be no, we don't get both functions defined until we hit right here. And if you look, we have an X value for now, both the G of X and the F of X. So we don't start getting functions until -3 and then we're going to go, when do we stop having both of them? True, right here. So 1234, if we look at -3 to four, we can see that every X value in between -3 and four will intersect both the blue and the purple lines. So they're defined. The domain of F + g of X is going to be -3 less than or equal to X, less than or equal to four. If we wanted to add 2 functions, let's say F + g at one. What we're going to do here is we're going to remember that that's really just saying F of 1 + g of one. And if we look at F of one, we come over here to our graph and we say, well, what's the value when X is one for F equation? And our value, our Y value out when our X is one is going to be two. So we're going to have two. And when we look at our G when our X is one, we get a three. So 2 + 3 is going to give us 5 S F + g of one is going to be 2 + 3 or five. We can do the same thing for multiplication and also division F * g of 0. That's really saying what's F is 0 * g of 0. So looking at this graph now with 0AS our X value, our F equation is going to give us a two and our G equation is going to give us 1234. So 2 * 4 is 8. The last example using the graph is going to be F of -2 / g of -2. Looking at the graph again, where is F of -2? Right over here. So F of -2 was going to be two and G of -2 is 23456, so this is going to be 1/3. That's how we would read a graph to find algebraic composition or combinations of two functions. The next thing we're going to look at is if we're trying to find it given algebra. So if I want F + g of four, what that's really asking me to do is to find F of 4 + g of four. So we're going to take that F equation and our input or our domain is going to be 4. So we're going to put four in every time we see the X. If we look at the G of four, we're going to stick 4 into the G equation every time we see the X. And then we're just going to combine. So 12 - 5 + 16 + 7, that's going to give us 30 if we had asked to find, if we had been asked to find F + g of X instead, we'd take the F of X equation and the G of X equation. Well, the F of X equation three X - 5, the G of X equation X ^2 + 7. So when we combine like terms here, we get X ^2 + 3 X +2. That would be what F + g of X is. Now we could actually do a check if we wanted to. If we wanted to use that F + g of X equation and now put in the four, it better give us out 30. So does 4 ^2 + 3 * 4 + 2 give us out 3016 + 12 + 2? That's a nice way to check it where you can put in the pieces into the original functions or you can just find the composition actually the combination and put the four into there. The last thing we want to look at is domains and the domain here of F of X equal four X + 5. What it's really asking us is what values can we put in for X where it's still defined? And this first one's kind of tricky because everything I put into X will be defined. So this one's going to be all real numbers, sometimes a capital R with two lines on it, or just all real numbers. The next one, we can never ever ever have a denominator that equals 0. So we know that three X -, 2 can't equal 0 3X can't equal 2X can't equal 2/3. Now the last part gets a little tricky. We're going to have F of X / g of X. So we have four X + 5 / 6 X divided by three X -, 2. Now we know the G of X equation can't have X being 2/3. What we need to think about is division is really the same thing as multiplying by its reciprocal. The second term is reciprocal. So now when we look at this and we can think about four X + 5 / 1, we can never have zero in the denominator again. So the 6X can never ever be zero. So X can't be 0. So for this last F / g of X, X can never be zero or two thirds. You guys have been awesome, thanks a lot.