 This video is going to talk about functions, so first we need to define a function, that's why I've called it the worksheet that you have. The function is a set of ordered pair where you use the think of x, y, and the first component, like the x, is paired with only one of the second component. So if I had one, two, and one, three, that couldn't be a function because one was going with two and three. It can only go to one. The domain is all the possible x values of a function, and if we want to do this from a graph, which we will do in a little bit, you can just use the vertical line test. If your vertical line crosses your graph more than one place, that means that the x on that line has more than one y value. And then the range is the possible y values of a function, and this is just determined by any horizontal boundary kind of line. Like if you have a graph that tends toward the x axis, then it would get close to zero, but anything above, we'd have a boundary line there. Now just reviewing interval notation. Brackets are used to say that the boundary is included, and parentheses are used so they're not included. Infinities always have parentheses. Always write the smallest value to the largest value. You can have combinations of intervals, so you use the union sign, which just looks like a capital U. Down here, it shows you the example. I just want to show you that when we're in web work, they like to use infinity as INF, so I wrote it that way for you. So you have negative infinity to negative five, and you have four to infinity. That's what all that means. So let's get rid of that, and go see if we can determine some things about our graphs here. We want to know if it's a function, and then we want to determine the domain and range. And then we also want to use function notation and evaluate these two things. So vertical line test here, if I drew vertical line anywhere, would show me that this is not a function, because that x value has two y values that go with it. So it's not a function. And then we want to know what the domain is. All the possible x is how far to the left, how far to the right. And I usually use a capital D like this for meaning domain, and then its x is an element of what interval. Well, it goes from, it looks like, negative three over to positive three, and it doesn't go any further than that. That circle is no wider than that. But it includes those two points. So we say bracket smallest value, negative three to three, the greatest value, and it includes three. And the range is going to be a y as an element of what interval. And it looks like it goes up to one, two, three, four, and it goes down to one, two, negative three. And it includes, it hits that point, so it goes from negative three to four. Finally, we want to evaluate. So f of zero, this means x is zero, and we want to find y. So if x is zero, we're right here, then we find out that y actually has two different points. It has this point up here, which is four, and it has this point down here, which looks like negative three. So we would say that y is four and negative three. It's both those answers. Next one, we do vertical line tests. I've watched my pen going all over this graph. It only hits it once. So this is a function. Instead of writing the word out, I'll just put f for function. The domain for this one, x is an element of, and then it looks like it starts here. Notice there's not an arrowhead on that, so that would be negative four. But then it goes on forever, because the zero means it's going to keep going up and out. So we would say that it starts at four, and it's a closed circle, negative four. So we bracket negative four, and then it's going to go to infinity, and I think I'll write it as inf. And then the range here, y is an element of, well, the lowest point is that point where it started, and that was negative four as well, and it starts there, and it goes up forever. So it also goes to infinity with a parenthesis. I didn't finish this top problem up here. f of x is equal to four. That tells us that y equal four. So we have to find out what x is when y equal four. So y equal four is up here, and there's only one point, and that would be zero. x is zero. That's when y is four. Down here, again, f of zero, that means that x is zero, so we have to find y. And when we have zero here on the x, we also have zero for the y, and there's only one. And then f of x equal four. It tells us that y equal four, and x is equal two. When we come over here to four, one, two, three, four, and then we go up to our graph. It looks like it's also four. So it would be four there. So now we want to just look at function notation and evaluate. Remember that that means that this negative six is actually x. So wherever I see x, I'll replace it with negative six. So I've got x squared, so it's negative six squared, and then plus two times that negative six. So negative six squared is going to be 36, and two times negative six is going to be negative 12, and 36 minus 12 will give us 24. Now let's try it with one-half. Doing the same thing. One-half being squared plus two times one-half. Well, one-half squared is one-fourth, or point two-five, plus two times one-half would be one. So we actually end up with one point two-five. I don't care if you use the fraction of the decimal. That was a nice decimal, so I went ahead and just used the point two-five. Now let's keep going, but we're not using numbers this time, but we don't do anything different. Whatever's inside the parentheses goes in for the x's. We've got a c, and we've got a square of that, plus two times c, and that's really all you can do. c squared plus two c, if you'd rather. I'd take either one. One more. So now we have c plus one quantity squared plus two times c plus one. c plus one squared. Remember the shortcut? It's the first term squared, plus twice the product. So c times one would be c times two would be two c, and then square the last. So one squared would be one. And then if I distribute here, I get two c plus two, and then just simplify. c squared, two c plus two c is going to be plus four c, and one plus two will be plus three. So now we have to talk about domain of a function, when it's written in function notation. So how do we do that? Well, first thing you need to do is think about what do we know about square roots? We know that the square root, this value underneath the square root, then a has to be greater than or equal to zero, or else we don't have a real number. So here we're going to say that five a minus two has to be greater than or equal to zero. So five a is greater than or equal to two. Remember adding or subtracting doesn't change the inequality sign. And we're going to divide by a positive five, so that doesn't change the inequality sign. So we have a is greater than or equal to two-fifths. Now if I want to write that in interval notation, which is what it says, it means that this is the smallest value that I can have, or anything bigger. So it's greater than or equal to, so that means a bracket. The smallest value is that two-fifths. And then the largest value is going to be infinity, with a parenthesis. Now let's look at this next one. We've got a square root again, so that we know the five over x minus two has to be greater than or equal to zero. And we also know that x minus two, since it's in the bottom of a fraction, the denominator can't be zero. So we know that x cannot be, if we let x be two, that will give us zero. So x cannot be two. The only thing that we have to worry about here is what's going to be in our denominator. So we would say that x minus two has to be greater than or equal to zero, because that numerator is already positive. It's not going to change signs. So x has got to be greater than or equal to two, but it can't be equal to two. So in this case we would say that x, I forgot to write this last time, is an element of not including two up to infinity. We have t plus one got to be greater than or equal to zero. And we know that three t plus two cannot equal zero. Those are the two things that we really have to take into consideration here. So if t plus one is greater than or equal to zero, then t has got to be greater than or equal to negative one. Okay. Now three t plus two would be equal to three. Then three t can't be equal to negative two. And t can't be equal to negative two-thirds. Negative two-thirds is bigger than negative one. It's closer to zero. It's closer to the right. So we say it starts, it can be equal to, so it starts at negative one. And then it will go up to that negative two-thirds. But it can't include it, because we're saying it cannot be that. Then we can have anything on the other side of it, because that's still greater than negative one. So we have to again, parentheses are negative two-thirds, and then go to infinity.