In today's video, or set of videos actually, we're going to talk about functions and function notations. But before we can even get to functions, we need to start at maybe a little more basic step. What is the building block of a function?
Well, the building block of a function is actually a relationship. So what is a relationship? Well, okay, what is a mathematical relationship? We all know what relationships are, right?
In mathematics, when we talk about a relationship, the most basic form of a relationship is actually just a single point. Just a single x, y forms all of the relationships that we talk about in mathematics. Now, a single point, probably not that interesting.
Alright, you know, if we had the point 3, negative 1, you know, it's a point and it's a relationship. It's a relationship between the x that is 3 and the y that's negative 1. There is this relationship between the two of them, but it's really not all that exciting. We can dress it up a little bit. We can change it a little bit.
We could even write it as an equation. But notice that when we have an x here, it's in relationship with some y. It creates just a point. Now, in the case of an equation, it usually creates an infinite number of points, but it still is that basic unit of relationship, those points. That's all we're interested in.
Now, when we talk about points, we do need to have a little bit of a vocabulary that we talk about. So let's take a look at a little larger relationship. Not as exciting as an equation, but a little bit better than a point. Here we've got...
a collection of four points you'll notice that when we talk about relationships and a collection of points we'll use these to start and end our relationship those are called curly braces and they tell you that everything on the inside goes together right so notice that we've got a whole bunch of X's and we've got a whole bunch of Y's and frequently it's important for us to be able to talk about the X's and the Y's so we give them specific names. So when we talk about the x values, we usually refer to that as the domain, and when we talk about the y values, we usually refer to that as the range. Although sometimes if your textbook or the person you're talking to is a little bit older, you may hear them referred to it as the image. And while this term is falling out of favor in mathematics and the sciences, I want to tell you it because, you know, they're still in use in some textbooks out there. So the range and the image are the exact same thing, right?
So here we've got our relationship, and someone might be wild and crazy and ask us for, what is the domain of this relationship? Well, we come in here and we look out and we say, oh, well, the domain is one, two, three, and four. All I did was pull all of the x values from my domain, as opposed to someone who might say, what is the range or the image, but we'll use range here.
And we'd pull all the y's and that would be two, three, four, and five. And that's really all we have as far as what domains and ranges are. All right, but As we start to get into equations, this becomes a little more interesting.
Let's look back at our original equation that we wrote up here in purple. y is equal to 3x plus 1. And we see that really, truly, there are no issues ever in the domain. There isn't a number that you can think of in the real number system. that we can put in here. So, how we would express the domain is we would go from negative infinity to positive infinity.
Or, if you prefer, sometimes you'll see it written as a script R for all of the real numbers. Now, the domain is usually a little bit easier to calculate than the range, and it helps to know what the picture of what we're working with when we talk about the range, it helps us find the range easier. For the most part, we'll concentrate on the domain because we're usually interested in what can go in and what can't go in.
So how do we tell when we have domain issues? Let's take a look at... where a domain might fail and let's take a look at the square root of X plus 2 all right now here not all numbers can go in there there are some numbers that you can think of then we plug in there they cause some problems so let's take a look at what those might be suppose for just one second that we allowed X to be equal to negative 6 all right so we plug that in there we get the square root of negative 6 plus 2 which is the square root of negative 4 and right away we should see we have some problems because there are not two numbers that are exactly the same that you can multiply by each other to get negative 4. There are no real numbers that we can do that with. There are some other numbers which we don't generally use, even in the calculus class that will make this happen, but for us... the real number system, there's no numbers that will multiply to give us a negative.
So, where you have to watch out for domain issues is when we're underneath a square root or an even power root. In this case... x plus 2 must be greater than or equal to 0 or x must be greater than or equal to negative 2. And the way we would write that for the domain is we would go negative, oh, no, I'm sorry. negative infinity.
Negative 2 is the smallest one we can go to and positive infinity is where we're headed to. Now you'll notice Mr. Bowen has used a little bit different symbology here. He used a square square brace on the left and a parentheses on the right and this was not by mistake when we look at the square brace this is speaking very clearly to all of us and it says include negative 2 as opposed to the parentheses which means exclude We want to exclude infinity.
Ooh, that was a terrible infinity. That's much better. Alright? So, the idea of inclusion and exclusion.
Inclusion means you can use that number. Exclude means you can get really, really close to it. but you can't use that number.
Now, infinity always is excluded. Infinity is really not really a number. It's more like a direction. If I told you to head east from where you are right now, you'd be like, You could keep walking east, but how would you ever know when you got to east? You wouldn't, because it's a direction.
And infinity is the same. It's a direction. It tells you to either go one direction on the number line, or it tells you to go the other direction on the number line.
So it's just telling you which way on the number line you're supposed to be going. And because it's a direction, we don't include it. All right?
One other place that we might run into issues is that that, for example, we might see x in the denominator of an equation. All right. And for the most part, 3 divided by x plus 1 is okay, except when x is equal to negative 1. Because when we plug negative 1 in there, we get a 0 in the denominator. And we've learned long ago that zeros in the denominator create undefined numbers.
It's not that there isn't an answer, it's just mathematically currently we're undefined. We don't know what it is. So we would come in here and we would say, oh, the domain for this.
equation is negative infinity we won't include it will come up to negative 1 and this time we won't include it then we'll go from negative 1 up to positive infinity because those are okay too and when we have more than one domain, we put them together with this U or a union symbol. For the most part, we're really interested simply in looking at our domains, mainly because we don't want to plug an X in there. that doesn't belong and cause things to fail.
That's what we're really looking at. On a whole, remember, we're talking about relationships and functions. This is the beginning. This is the building block for all functions.
We'll talk about functions in our next video.