 All right, so in this final video for lecture two in our series, we're gonna finally start thinking about doing some algebra with functions. Now, be cautious here, the functions themselves are not going to be algebraic. Not yet, so that'll be coming very, very soon. But even if we have a function that is graphical, even if we have a function that's numerical, we can do algebra with those functions. For example, it makes sense to add together functions F plus G because if our functions are numerical that is they input a number, they output a number, we can add those numbers together and thus we can create a way of adding the function. So when we add functions F plus G, we define that to be the sum of their images, the sum of the output of the function. We add together F of X with G of X, right? Because remember F of X is just the Y coordinate associated to the function F when you plug in X. And G of X is, it's not the function, it's the Y coordinate connected to X when you stick X inside of the G machine. So we're gonna define F plus G evaluated X as F of X plus G of X. And we're gonna do that for all four arithmetic operations. F minus G of X is defined to be F of X minus G of X. F times G evaluated X is evaluated as F of X times G of X. And then F divided by G of X is defined to be F of X divided by G of X. So we can do arithmetic on functions by doing arithmetic on the output numbers of those functions. Now there are some domain issues we have to be careful about. In order to define F plus G, F of X has to be defined and G of X has to be defined. So the only way we can find F plus G of X is that F of X has to be a number and G of X has to be a number. So X has to be inside the domain of F and it has to be inside the domain of G. And so if we have those considerations, the domain of F plus G will be the intersection. That's what the symbol right here means intersection. It's like an upside down union symbol. The intersection of the domain of F with the domain of G will be the domain of F plus G. That is if F of X is defined and G of X is defined, then we can define F plus G of X. We do the same thing for subtraction. The domain of F minus G will be the intersection of domains of F and G. The domain of F times G will be the intersection of domains of F and G, right? Those kind of make sense. That's what we do for division as well. For F of X divided by G of X to be well-defined, F of X has to be defined, G of X has to be defined. But there is one other little hiccup we have to be aware of when we talk about division here. We have to make sure that the denominator G of X, we have to make sure that thing is not zero. And so we have to also stipulate that G of X is not zero for that choice of X. Otherwise the thing would be undefined. Dividing by zero is not gonna produce a number here. And so let's look at an example of this if we focus on just some two functions that are expressed numerically. So we have one function F whose domain is gonna be the numbers one, two, three, four, five, six. And the associated y-coordinates are these numbers which we can read from the table. We also have a function G illustrated right here. Its domain will be zero, one, two, three, four, five. And the associated y-coordinates will be these six numbers you can see right there. So if we wanna define F plus G, the first thing to think about is actually the domain, right? What's the domain of this thing? The domain of F plus G, like we saw in the previous slide, this is gonna be all the numbers for which F and G are simultaneously defined. So notice that F is defined, it's defined one through six but G is defined zero through five. So you'll notice zero is defined for G but it's not defined for F. One, two, three, four are good but six is defined for F but not defined for G. So the domain of our function is gonna be one, two, three, four, and five. We can't define F plus G at one, sorry, we can do that one. We can't do F plus G at zero because F of zero is undefined. We can't do F plus G at six because G of six is undefined. So we do get one through five as the domain of this function. And so then how do we identify the number that goes in here? We're just gonna, so for F plus G of one, we're gonna take negative four plus four and that's equal to zero. That's what we get right there. So to do F plus G of two, we're gonna take zero plus negative two which is negative two. So that gives us F plus G of two. If we wanna do three, we're gonna take F of three which is negative two plus G of three which is zero and that's gonna add to negative two. That's the sum in that situation. F of four is one, G of four is negative seven, the sum will be negative six and that's what we define F plus G at four. And then finally, F of five is three, G of five is 11, the sum would be 14. So F plus G of five is equal to 14. That's all there is to it. You have to just add the Y coordinates together if both of them are simultaneously defined. Now the difference of F and G would be computed similarly. The product of F and G would be done similarly. I wanna mention next now the quotient of F and G. So the quote to find the domain of F divided by G, right? That one's a little bit different because we have to look for points that are simultaneously defined. We already did that but we also cannot let the denominator equals zero and you'll notice that when X equals three, G of three is zero. So three we have to remove from the domain. And so the domain will only be points one, two, four and five. Notice three is removed from consideration here. Now it's okay that F of two is equal to zero because in that situation, F of two would be zero divided by negative two, G of two is negative two. So F divided by G at two would be zero divided by two and zero divided by two is, it's defined. This is just zero, right? If the numerator is zero, as long as the denominator's not zero that'll just be zero itself. So F divided by G at two is defined. And then the other numbers will be handled similarly, right? If you take for one, you'll take negative four divided by four, that'll give you a negative one. If you take one divided by negative seven, that gives you negative one-sevenths, you often get fractions when you divide. When you do five, F of five is three, G of five is 11. So F divided by G at five would be three-elevenths and that's how we compute it. And so there's not much more to it. Just make sure that we never divide by zero and we only do the operations when both functions are defined on their own. And then we can add, subtract, multiply and divide functions. This can be done graphically as well. You just look up the points on the graph and in the next lecture, because this is the end of lecture two, we'll explore how to start doing these function evaluations and other examinations of functions for the algebraic functions, which we'll spend most of our time with in this course.