WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about the idea of a function.
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Functions are extremely important to mathematics: you have certainly encountered them before.
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But you might not have fully understood how they work and what they are doing.
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This lesson is here to give us a clear understanding of what it means for something to be a function, and how functions work.
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Since functions are so important, they are going to come up in every single lesson you learn about in this course.
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And they are going to come up in every single concept you talk about in calculus.
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And they are going to keep coming up, as long as you are studying math.
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Make sure you watch this entire lesson; it is so important to have a good, grounded, fundamental concept of what a function is,
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because it is going to keep getting used in everything that we talk about.
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This is probably the single most important lesson of this entire course,
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because so many later ideas are going to talk about functions.
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Also, it would help to have watched the previous lesson on sets, elements, and numbers,
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because we are going to be talking about how sets are connected to functions.
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So if you haven't done that, I would recommend that you go and watch that one first,
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because it will help explain a lot of what we are talking about here, because functions are relying on the idea of sets.
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All right, let's jump into it: what is a function?
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A function is a relation between two sets: a first set and a second set.
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For each element from the first set, the function assigns precisely one element in the second set.
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So, we will point at some element in the first set, and it will say, "Here is an element from the second set."
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Point at another element from the first set, and it will tell us, "Here is some element from the second set."
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That is the idea of a function; here is a visual example for it.
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We could have something where all of the squares are the first kind--it is our first set--and all of the round things on this side are our second set.
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So, second would be the second column, and first set would be the first column.
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We could have news get put onto paper; we say that news, the function, gives us paper.
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We say that cheese, the function, gives us burger; we say that good, the function, gives us bye.
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We say that sand, the function, gives us paper; we say that bubble, the function, gives us gum.
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So, there are only five elements in our first set, and only four elements in our second set.
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But this is a perfectly reasonable function: newspaper, cheeseburger, goodbye, sandpaper, bubblegum.
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The only one you might be wondering about is..."Wait, news goes to paper and sand goes to paper."
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There is no problem with that: we only said that the function has to give us something when we point at something in the first one.
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We never said that it has to be a different thing, every single thing that we point to; it just has to give us something for it.
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That is what we have here: we have something where everything that we call out on the first side...
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we call out news, and in turn, it responds by telling us paper.
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We call out good, and in return, it says bye; that is how it is working here with this function.
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Here is a non-example: in this one, we say tree, but the function gives us four different possibilities.
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Sometimes it gives out maple; but other times it gives out oak; but other times it gives out apple; but other times it gives out pine.
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And then fruit--if we go to fruit, it sometimes gives out apple, and sometimes it gives out grape.
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This isn't allowed, because it is only allowed to give one response to a given input.
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We tell it one element from our first set; it can only tell us one element from the second set.
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It is not allowed to give us a whole bunch of different choices to pick and choose from.
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Sometimes it is going to be maple; sometimes it is going to be oak; sometimes it is going to be pine.
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No; it has to be one thing, and one thing only.
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That is what it requires to be a function; so this is not an example--this is not allowed, because we can't have it be multiple things coming out of this.
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It has to only be that one input will only give us one output.
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And as long as we keep putting in that same input, it can only give us the same output.
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Just like variables, it is useful to name functions with a symbol; so let's talk about how notation works here.
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Most often, the symbol we will use to talk about a function is f; but sometimes
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we are also going to use g, h, or whatever else will make sense, depending on the context.
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But often, we are going to end up seeing f.
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If we want to talk about what f assigns to some input x, if x is the element in our first set,
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if that is what we call the element in our first set that we use f on, then it will be assigned to "f of x,"
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f acting on x--what f gives out when given x; so the first symbol is the name of the function that we are using;
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then, the second symbol, in parentheses, is what the function is acting on.
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So, f--the name of our function--is acting on x; and then, that whole thing together is f(x); f(x) is the name of what comes out of it.
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So, f is the name of what is doing the acting; x (or whatever is in the parentheses)...the first symbol was the name
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of whatever is doing the acting; the thing inside of the parentheses is the name of what is being acted on;
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and then, the whole thing taken together is where we are when we use the function on that element--
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what we get output to where we come to.
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Now, there could be a little bit of confusion about f(x), because it is f, parenthesis, x.
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And we know that parentheses...if I wrote 2(3), that would mean 2 times 3, right?
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So, we might think f times x; but we are going to know from context that f is a function, and not something that we multiply.
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So, when f is a function, we don't have to worry about using multiplication, if it is f on some element.
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It is always going to be f of that element, never f times, unless we are talking about that explicitly.
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But if it is just in parentheses, it is not going to be multiplication.
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So, when you see parentheses, and it is a function, it isn't implying multiplication, like when we are dealing with numbers.
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If we want to express what sets the function acts on, we can write f:a→b.
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What this is: it is "f goes from a to b"; it takes elements from a, our first set; and then it assigns them elements from b.
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Normally, it won't be necessary for us in this course (and probably for the next couple of years)--
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it won't be necessary to name the sets that our function is working on.
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But why that is, we will discuss later: it is going to be pretty simple, but we will discuss it later when we get to it.
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There are a lot of metaphors that we can use to help us understand what is going on in a function.
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Here are three metaphors to help us understand what happens when f takes things from a and goes to b.
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Our first idea is transformation: the function transforms elements from one set into another.
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It takes an element x, contained in a, and then it transforms it into an element in b, which we call f(x), or f acting on x.
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f(x) is what it has been transformed into; that is what it is after the transformation.
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Now, from problem to problem, the rules for transformation will usually change as we use different functions.
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One function is generally going to have a different set of rules for how its function works than another function.
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But if we are using the same function--if we are in the same problem, using the same function--
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the rules never change: if we put in the same x, we will always get the same f(x) as our result.
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The rules for how the transformation works are always the same.
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So, if the same thing goes in, the same thing always comes out.
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Another way we can look at it is a map: it tells us how to get from one set to another set.
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It is sort of a guide, directions for how to get from one place to another place.
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Of course, if we start at a different starting location, a different starting place,
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different elements in a, we might end up at a different destination--different elements in b.
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If I say, "Go 100 kilometers north," you are going to end up in totally different places
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if you start in Mexico, if you start in California, if you start in England, if you start in South Africa, or if you start in Japan.
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Each one of these places...if you start in Egypt...is going to end up going to a totally different place, even though they are all still the same direction.
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You are still doing the same thing; you are still going 100 kilometers north in all of these cases.
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But because you started in a different place, you end up in a different place.
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So, a different starting place, a different element that we are acting on, a different element that we are mapping,
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will normally cause us to have a different destination--a different place that we land on.
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The math itself, though, never changes: if we start at the same place, we always arrive at the same destination.
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So, if we start in San Jose, California, and then we go 100 kilometers to the north...I actually have no idea where that is.
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But we will be 100 kilometers north of San Jose.
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And then, if we start in San Jose again on another day, and we go 100 kilometers north, we are going to end up being in the exact same place.
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And if we go to San Jose, and then we go 100 kilometers north again, we are going to end up being in the exact same place.
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And people are probably going to wonder, "Why does this person keep showing up here?"
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And that is because we are following the same map.
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The directions, the transformation that the map gives us, the way we go, isn't going to change each time.
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It only changes when we start from a new place.
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Finally, one last way to visualize it is the idea of a machine.
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We can visualize a function as a machine that eats elements from a, and it produces elements from b.
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What it produces depends on what it eats, but the machine is reliable: if it eats the same thing, it always produces the same output.
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For example, if we have x right here, and we push it into our machine, f, it goes into the machine;
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then the machine works on it and crunches it, crunches it, crunches it; and it gives out f(x).
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So, we are going from the set a to the set b.
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Now, one thing about the machine is that it is perfectly reliable; the machine is reliable.
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If it eats the same thing, it produces the same output.
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If we put in x, it will always give out f(x); so the first time we put in x, it gives out f(x); the second time, f(x); the third time, f(x); the fiftieth time, f(x).
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Just like when we started in San Jose, and we went 100 kilometers north, each time we always ended up coming to the same place;
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you put the same thing into the machine; the same thing comes out.
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This idea is so important; we are going to talk about it really explicitly.
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We have said this one way or another for all of our different ways of thinking about functions.
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But it is so important--it is such an important characteristic of functions--we want to make sure that we know it.
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If we put the same input into a function, it will always produce the same output.
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Now, the input and the output could be totally different; the input is not necessarily going to be where we show up in the output.
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You start in San Jose, and then you show up in some farmer's field 100 kilometers to the north.
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But you are going to come out to that same farmer's field each time, because you are showing up at the same location.
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So, for a function to make sense and be well-defined, for it to work, its rules must never change.
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For example, if f(2), f acting on 2, gives out 7; if f(2) equals 7 the first time, then f(2) = 7 the second time; and f(2) = 7 every time.
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No matter how many times f operates on 2, no matter what, it is always going to give out the same thing.
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That is what it means to be a function: your rules don't change when you are going on the same thing.
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You work on one element the same way each time; you always map it; you always transform it; you always assign it to the same place.
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Here is something that is not a function: g(cat) = fur, g(cat) = whiskers, g(cat) = quiet.
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This can't be a function, because we have three totally different destinations when we plug in cat.
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And what determines whether we go to fur, whiskers, or quiet?
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There is no reason why we should use one set of rules or another set of rules, so it is not a function.
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There is no reliability here; we don't know, when we plug in cat, if we are going to go to fur, whiskers, or quiet.
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So, it is not a function; but we could have a function that was h(fur) goes to cat, h(whiskers) goes to cat, h(quiet) goes to cat.
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It is not that there is a problem with having us land on the same place.
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No matter what we put in, the function could give out cat: it doesn't matter,
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as long as the first thing, the first set we are coming from, can't split as it comes out.
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We can land on the same place, but we can't be coming from the same place and go to two different locations.
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We always have to follow one rule; because we are following one rule, we can't land on two different things.
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Let's look at a non-numerical example: before we start telling you about how functions work on numbers,
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let's consider an example of one that works on something totally not about numbers.
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Let's think about a function that gives initials: we will define...f is going from names spelled with the Roman alphabet
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(names like Vincent or John, not things that are spelled with characters that we can't express in the Roman alphabet),
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and it is going to go to letters from the Roman alphabet.
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So, f(x) equals the first letter of x; now, if we say, "Wait, we know that the first letter of x is x!"--
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yes, but what we are talking about is names: x is a placeholder, remember?
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We talked about variables: the idea of a variable is that it is a placeholder.
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So, x is just sort of keeping the spot warm, until later, when we put in the name.
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So, if we decide to put Vincent into the function, then this x on the left side tells us where to put Vincent on the right side.
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So, Vincent will come in here on the right side, as well.
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We will have Vincent go on the left, and Vincent will go on the right.
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f(Vincent) would be V: we cut it off just to the first letter.
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f(Nicole) would put out N; f(Padma) would give out P; f(Victor) would give out V; f(Takashi) would give out T.
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Whatever we put in, it will give out just that single letter.
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So, if we were to turn this into a diagram, we could have Vincent here, Nicole next to Vincent, Padma, Victor, and then finally Takashi.
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And so, this is where we are coming from; and then, we are going to letters.
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So, we have V and N and P and T...and let's put in another letter, like...say S and Q.
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Vincent gets mapped to V; Nicole, by this function, gets mapped; Padma gets mapped to P; Victor also gets mapped to V.
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Takashi gets mapped to T; but do S and Q get used? Not for this set of names.
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Maybe if we put in Susan, or we put in...there has to be a name with Q that I don't know...
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let's pretend that the name is simply Queen...I am sure that there is a name...a really weird spelling of the name Cory?...
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there is a name out there that is spelled with a Q; I just don't know it immediately.
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So, there is something out there that can fill up that S, and that can fill up that Q; we just don't have it in what we are looking at so far.
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So, there might be other things that we are not hitting on the right;
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but everything that we have on the left is what is getting mapped to things on the right.
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So, the functions we use...of course, it is no surprise; this is math--we are probably going to be talking about numbers.
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So, it shouldn't come as a surprise; we are going to concentrate on using these functions with numbers.
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Functions, as we just saw, can be used for lots of things; but we will focus on functions and the real numbers.
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Unless we are told otherwise, we will assume that every function takes in real numbers and outputs real numbers.
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That is to say, f is taking in reals and then giving out reals.
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OK, so when we are given a function, we will usually be told what its rule is--how it maps inputs to outputs.
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So, for example, if f(x) = x² + 3, its rule is "Square the input," since x is our input;
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then, what we do is...we first square the input, and then we add 3; square the input and then add 3.
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That is its rule; that is how it works.
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Notice that x acts as a placeholder; just like it did with the names, it acts as a placeholder.
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It is not that x is really the thing we are worried about being acted on.
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It is just telling us what is going to happen to whatever we plug into this function.
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If we plug in 3, what will happen to 3?
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If we plug in 50, what will happen to 50?
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If we plug in smiley-face, what will happen to smiley-face?
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x is just there to sort of keep a spot warm: it is telling us, "Here is the place; things will go into this place."
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And things will go into this place, wherever I show up on the right side, as well.
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If we want to use a function, if we want to evaluate a function at a specific value, we just apply this rule to whatever our input value is.
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In practice, this turns out to actually be really simple.
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Usually, we are given a formula for each function; so we just follow the method of substitution.
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Remember, we take whatever we are substituting in; we wrap it in parentheses; and then we see what we get.
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For example, our function is f(x) = x² + 3; then, to find f(7), we just plug in.
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7 is what we are plugging in; so we have 7 in this spot, and a 7 will go in here.
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We wrap that in parentheses, just in case; in this case, we don't have to, but we will see why it is useful to always remember to wrap it in parentheses.
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7² + 3...7² is 49; 49 + 3...we get 52.
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If we want to look at a slightly more complex example, though, we see why it is so important to wrap your substitutions in parentheses.
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If we consider a slightly more complex input, like a + 7, then we have to have it in parentheses,
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because it is not just the a that gets squared; it is not just the 7 that gets squared; it is all of that thing that went in.
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All of that thing is both the a and the + 7; it is (a + 7); it is that whole number combined.
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It is not a² + 7; it is not a + 7²; it is (a + 7), the whole thing squared; and then, plus 3.
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A good way to see the behavior of a function is by creating a table of values; sometimes we call it a T-table, because it has the shape of a T.
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On one side, we have input values, while the other side shows us what the function outputs when given that input.
00:17:50.700 --> 00:17:56.100
So normally, the left side will be our input value, and the right side will be our output value.
00:17:56.100 --> 00:18:00.500
So, for example, if f(x) = x² + 3, then we can give out a bunch of values for it.
00:18:00.500 --> 00:18:06.100
So, if we want to figure out what happens to f(-2), we just follow the normal thing.
00:18:06.100 --> 00:18:15.500
f(-2), so we plug it in...(-2)² + 3...we get 4 + 3; we get 7, and that 7 shows up here.
00:18:15.500 --> 00:18:22.900
If we want to figure out what f(-1) is, we do the exact same thing: (-1)² + 3, 1 + 3, and 4.
00:18:22.900 --> 00:18:26.500
And that 4 shows up here; and so on, and so forth.
00:18:26.500 --> 00:18:32.400
We just plug in, based on this rule...whatever the rule we have been given...we plug in whatever our input is,
00:18:32.400 --> 00:18:35.700
whatever the thing on the left is, any of these numbers.
00:18:35.700 --> 00:18:42.400
And then, once we figure out what this number is here, we figure out, we evaluate, and we get what its corresponding value is on the right side.
00:18:42.400 --> 00:18:45.800
And we write that in, and that is how we make a table of values.
00:18:45.800 --> 00:18:55.600
Having this table is often a very useful way to quickly analyze and see what is happening in a function over a large range of possible inputs.
00:18:55.600 --> 00:19:01.200
Domain: the **domain** is the set of all inputs that the function can accept.
00:19:01.200 --> 00:19:06.100
The domain is what can go into the function: it is the inputs that we are allowed to use.
00:19:06.100 --> 00:19:09.700
It is what our machine can eat without breaking down.
00:19:09.700 --> 00:19:14.600
Well, we generally assume that all of ℝ can be used as inputs--all of the real numbers can be used as inputs.
00:19:14.600 --> 00:19:19.800
Sometimes, certain values will break our function; the output won't be able to be defined.
00:19:19.800 --> 00:19:25.600
Thus, our domain is normally going to be all of the real numbers, except those numbers that break our function.
00:19:25.600 --> 00:19:29.400
Occasionally, we might actually get things where we are going to be given an explicit domain--
00:19:29.400 --> 00:19:34.700
like just evaluate it from -3 to 3--and forget everything beyond those -3 and 3 values.
00:19:34.700 --> 00:19:39.600
But normally, we are going to assume all of ℝ, except those things that break our function.
00:19:39.600 --> 00:19:47.700
Let's see an example: if we had f(x) = 1/x, the function would be defined, as long as we don't divide by 0.
00:19:47.700 --> 00:19:56.000
If we have x = 0, though, then f(0) gets us 1/0.
00:19:56.000 --> 00:20:02.200
Are we allowed to do that? No--that is very bad; we cannot divide by 0.
00:20:02.200 --> 00:20:05.700
So, we are not defined there; everything else works, though.
00:20:05.700 --> 00:20:08.600
If we plug in anything that isn't a 0, it works out fine.
00:20:08.600 --> 00:20:14.300
So, everything is defined, as long as x is not 0; so our domain is all numbers, except 0.
00:20:14.300 --> 00:20:20.700
The domain of f, to show all numbers except 0, is everything from -∞ up to 0, not including the 0,
00:20:20.700 --> 00:20:25.000
and then union with everything from 0, not including the 0, to ∞.
00:20:25.000 --> 00:20:29.700
That is just another way of expressing all of the real numbers, with the exception of 0.
00:20:29.700 --> 00:20:34.700
Now, for now, we mostly only have to watch out for dividing by 0 and taking square roots of negative numbers.
00:20:34.700 --> 00:20:37.200
Those are the only two things we have to worry about breaking functions.
00:20:37.200 --> 00:20:43.300
However, you can't take the square root of a negative number, because what could you square that would still have a negative with it?
00:20:43.300 --> 00:20:47.500
Any number, squared, becomes positive; so we can't have the square root of a negative number,
00:20:47.500 --> 00:20:50.000
because it would be impossible to give me a number that you could square
00:20:50.000 --> 00:20:53.300
into making it negative--at least as far as the real numbers are concerned.
00:20:53.300 --> 00:20:57.400
Later on we will talk about the complex; but that is for later.
00:20:57.400 --> 00:21:01.300
Right now, we only really have to worry about dividing by 0 and taking square roots of negative numbers.
00:21:01.300 --> 00:21:04.300
Those are the things to watch for; that is where our domain will break down.
00:21:04.300 --> 00:21:09.100
Later in the course, we will have a little bit more to worry about; we will also have to worry about inverse trigonometric functions.
00:21:09.100 --> 00:21:14.100
Those are only defined over certain things; and also logarithms have some parts that they are not allowed to take, either.
00:21:14.100 --> 00:21:17.200
But right now, it is just dividing by 0 and taking square roots of negative numbers.
00:21:17.200 --> 00:21:21.100
And later on, much later in the course, after we see these ideas, we will have to think about them, as well,
00:21:21.100 --> 00:21:24.600
when we are thinking about the idea of what can go into a function.
00:21:24.600 --> 00:21:29.300
Domain is what goes in; **range** is what comes out.
00:21:29.300 --> 00:21:34.400
Range is the set of all possible outputs a function can assign, given some domain.
00:21:34.400 --> 00:21:42.200
With some domain to start with, these values are what is able to come out: the range is what can come out, given some domain.
00:21:42.200 --> 00:21:46.900
These values will always be in the real numbers, unless we are dealing with a set that isn't working in the reals.
00:21:46.900 --> 00:21:49.300
But they don't necessarily cover all of the reals.
00:21:49.300 --> 00:21:53.600
For example, that function we were working with before, f(x) = x² + 3:
00:21:53.600 --> 00:21:58.900
the lowest value that f can output is 3, because the smallest number we can make with x²...
00:21:58.900 --> 00:22:03.600
well, x² always has to be greater than or equal to 0, because there is no number
00:22:03.600 --> 00:22:07.400
that we can plug into x and square that will cause it to become negative.
00:22:07.400 --> 00:22:15.500
The lowest we can get that down to is a 0, so the lowest we can make this whole thing is when this is a 0, plus 3; so the lowest possible output is 3.
00:22:15.500 --> 00:22:19.900
We can produce any value above 3 with x², though, so we can just keep going up and up.
00:22:19.900 --> 00:22:24.000
So, our range would be everything from 3, including 3, up until infinity.
00:22:24.000 --> 00:22:28.900
So, it is all of the reals from 3, including 3, and higher; great.
00:22:28.900 --> 00:22:32.800
If we want to look at an example that doesn't use numbers, we could talk about that initial function,
00:22:32.800 --> 00:22:37.000
that function that ate names and gave out first initials, from earlier in this lesson.
00:22:37.000 --> 00:22:42.600
In that case, if the domain is all names, then the range is all 26 letters of the Roman alphabet,
00:22:42.600 --> 00:22:51.700
even though I still can't think of any names that start with a Q...Queen...let's say Queen counts.
00:22:51.700 --> 00:22:59.500
OK, Queen Latifah, right?--it has to count; then, we can have that be the range--26 letters for the Roman alphabet.
00:22:59.500 --> 00:23:02.900
So, because if we are looking at all the names that could possibly exist...
00:23:02.900 --> 00:23:11.600
well, there is Albert; there is Bill; there is Charles; there is Doug; there is Elizabeth...and so on, and so on, and so forth.
00:23:11.600 --> 00:23:13.700
So, there is always something that will put that out.
00:23:13.700 --> 00:23:19.700
But if we restricted the domain to the five names that we saw earlier, Vincent, Nicole, Padma, Victor, and Takashi,
00:23:19.700 --> 00:23:24.800
then we only had four letters show up--we just had N, P, T, and V show up.
00:23:24.800 --> 00:23:29.900
So, in that case, if we restricted our domain to a smaller thing, our range would also shrink.
00:23:29.900 --> 00:23:32.400
So, the range depends on what our domain is.
00:23:32.400 --> 00:23:37.700
If we are looking at...normally we look at everything that can go into the function, and that is normally how we think of the domain.
00:23:37.700 --> 00:23:39.700
So, the range is everything that could come out.
00:23:39.700 --> 00:23:45.100
But sometimes, we will be given a more restricted domain, and we have to think in terms of that more restricted domain.
00:23:45.100 --> 00:23:47.000
All right, we are ready for some examples.
00:23:47.000 --> 00:23:50.600
First, there are nice, easy ones to get us warmed up to this idea of plugging in.
00:23:50.600 --> 00:23:53.800
If f(x) = 3x - 7, what is f(2)?
00:23:53.800 --> 00:24:06.800
We just plug in...if we use red for this...f(2)...we plug in 3; plug in that 2; minus 7...3 times 2 equals 6; 6 minus 7...so we get -1.
00:24:06.800 --> 00:24:15.200
Let's use blue for this one: if we have f(-4), then 3...we plug in that -4, minus 7.
00:24:15.200 --> 00:24:20.500
3 times -4 is -12; minus 7...we get -19.
00:24:20.500 --> 00:24:25.300
Oh, no! What if we have to use something that is variable? No problem.
00:24:25.300 --> 00:24:30.300
We still just follow the exact same rules: f(a)...well, what happened to x?
00:24:30.300 --> 00:24:39.000
It became 3x - 7, so now it is going to become 3a - 7, so we get 3a - 7.
00:24:39.000 --> 00:24:50.500
And what if we want to do b + 8? The same thing--f(b + 8) = 3(b + 8) - 7.
00:24:50.500 --> 00:24:55.500
So, we have to distribute; and notice how important it was that we put it in parentheses.
00:24:55.500 --> 00:25:00.900
If we had just plugged in this 3b + 8, that would be totally different than 3(b + 8).
00:25:00.900 --> 00:25:06.000
And that is what it really has to be, because it is everything in here that got plugged in, not just the b.
00:25:06.000 --> 00:25:09.400
The b and the 8 don't get to be separated now; they have to go in together.
00:25:09.400 --> 00:25:22.100
So, 3(b + 8) - 7...we would get 3b + 24 - 7, which is equal to 3b + 17.
00:25:22.100 --> 00:25:27.100
The next one: what if we wanted to fill in a table, g(z) = z² - 2z + 3?
00:25:27.100 --> 00:25:39.400
If we had to fill in this table, then we could do g(-1), (-1)² - 2(-1) + 3,
00:25:39.400 --> 00:25:50.700
equals 1 (-1 times -1 is 1), minus 2(-1), so plus 2, plus 3, equals 6; so we get 6 here.
00:25:50.700 --> 00:26:03.700
Next, g(0): 0² - 2(0) + 3...that simplifies to just 3, because of the 0's; they disappear.
00:26:03.700 --> 00:26:21.300
If we want to plug in g(1), then we get 1² - 2(1) + 3, so 1 - 2 + 3 comes out to 2.
00:26:21.300 --> 00:26:39.800
We plug in g(2); we get 2² - 2(2) + 3 = 4 - 4 + 3, which is 3.
00:26:39.800 --> 00:27:01.200
We plug in 10; we get 10² - 2(10) + 3; 10² is 100, minus 2 times 10 (is 20), plus 3 equals 83.
00:27:01.200 --> 00:27:05.200
There we go: so you just plug into the function exactly as you would to set up this table.
00:27:05.200 --> 00:27:14.400
You are told what your input is; and then, over on the right is your output, based on the rules of the function.
00:27:14.400 --> 00:27:24.800
The function gives us some rules, and so we plug in inputs like -1, and -1 goes through: (-1)² - 2(-1) + 3; we get 6.
00:27:24.800 --> 00:27:28.700
And that is what is going on when we are making a table of values.
00:27:28.700 --> 00:27:35.600
If h(x) = 2x² + bx + 3, and we know that h(3) = 15, what is b?
00:27:35.600 --> 00:27:40.200
So in this case, we are looking to figure out what b is.
00:27:40.200 --> 00:27:46.800
Now, we know that h(3) = 15; so we need to somehow use this to figure out b.
00:27:46.800 --> 00:27:52.600
So, we think, "I could plug in 3, and I would get something different than just 15."
00:27:52.600 --> 00:28:01.600
So, h(3), based on the rule, is 2(3²)...we are switching for where all of the x's show up;
00:28:01.600 --> 00:28:26.800
x here; x here; that is it; so 2(3²) + b(3) + 3...so we get 2(9) + 3b + 3, which is 18 + 3b + 3, or 3b + 21.
00:28:26.800 --> 00:28:35.400
Now, at this point, we say, "Right; I also know that h(3) is 15; well, this is still h(3), right?"
00:28:35.400 --> 00:28:44.800
So now, we put h(3) = 15, and we swap it out, and we get 15 must equal what we know h(3) is.
00:28:44.800 --> 00:28:50.500
We know that h(3) is equal to 3b + 21; and we also know that h(3) is equal to 15.
00:28:50.500 --> 00:28:58.800
So, since h(3) is two different things, but it is still just h(3), we know that they must be the same thing; otherwise there is no logic there, right?
00:28:58.800 --> 00:29:23.600
So, 15 = 3b + 21; we subtract 21 from both sides; we get -6 = 3b; we divide by 3 on both sides, and we get -2 = b.
00:29:23.600 --> 00:29:31.000
The next one: What is the domain and range of f(x) = 12 - √(x + 3)?
00:29:31.000 --> 00:29:54.000
Now, remember: domain is what can go in; range is what can come out.
00:29:54.000 --> 00:30:01.500
What we first one to do is figure out the domain first: what can go in without breaking this function?
00:30:01.500 --> 00:30:03.400
So, is there anything that can break in this function?
00:30:03.400 --> 00:30:11.800
We say, "Oh, right, the square root breaks when there is a negative inside."
00:30:11.800 --> 00:30:16.300
We can't take the square root of -1, because there is no number that you can give me--
00:30:16.300 --> 00:30:21.800
at least no real number that you can give me--that would square to give us -1.
00:30:21.800 --> 00:30:25.400
You give me any positive number; it comes up positive; you give me any negative number; it comes out positive.
00:30:25.400 --> 00:30:31.300
You give me 0; it comes out 0; so there is no number you can give me that will give out a negative number when squared.
00:30:31.300 --> 00:30:34.800
So, square root breaks when we are trying to put a negative inside of it.
00:30:34.800 --> 00:30:42.500
So, when will this break? √(x + 3) breaks when we have a negative inside.
00:30:42.500 --> 00:30:50.400
So, when is (x + 3) going to be negative? when x is less than -3.
00:30:50.400 --> 00:31:00.200
So, if x is less than -3, if x is more negative than -3, then this will be a negative value inside.
00:31:00.200 --> 00:31:03.500
If, for example, we use -4, then we will get the square root of -1.
00:31:03.500 --> 00:31:08.700
If we put in negative fifty billion, then we will get the square root of negative fifty billion plus 3, which would definitely still be negative.
00:31:08.700 --> 00:31:13.300
So, it only stops being a negative inside when we actually get to -3.
00:31:13.300 --> 00:31:19.700
-3 is an allowed value, because √(-3 + 3) would be √0; we do know the square root of 0--it is 0.
00:31:19.700 --> 00:31:28.400
So, the domain works for -3 and higher; everything is still reasonable higher than that.
00:31:28.400 --> 00:31:34.800
Our domain is going to include -3, and it is going to go for anything higher than that.
00:31:34.800 --> 00:31:47.500
So, that is our domain; if we want to figure out what the range is, then the question is, "What can f(x) put out?"
00:31:47.500 --> 00:32:00.600
So, notice: we have 12 - something; that something, √(x + 3)...square root can give out any number.
00:32:00.600 --> 00:32:08.100
If you put in √0, √1, √4, √9...you are going 0, 1, 2, 3, and you can make any number in between that.
00:32:08.100 --> 00:32:12.600
12 - something...what is the smallest that something could be?
00:32:12.600 --> 00:32:25.700
The smallest number that that something could be is 0; so that is smallest when √0 = 0.
00:32:25.700 --> 00:32:36.500
The biggest number we can get is 12; 12 is the highest number we can get, the largest number we can get out of this function.
00:32:36.500 --> 00:32:41.000
What is the smallest number we can get? Well, you can just keep giving me larger and larger x
00:32:41.000 --> 00:32:45.400
to make our square root a bigger and bigger negative number, on the whole.
00:32:45.400 --> 00:32:51.100
It would be minus larger and larger numbers; so 12 minus larger and larger numbers...we can keep going down.
00:32:51.100 --> 00:33:02.100
So, any number below 12 can be achieved, because we can just keep having the square root give out slightly larger and slightly larger numbers, which...
00:33:02.100 --> 00:33:05.800
Since we are subtracting by these larger and larger numbers, we will keep going down.
00:33:05.800 --> 00:33:17.300
So, any number below 12 can be achieved; so we have our range--it is going to be everything from the lowest possible,
00:33:17.300 --> 00:33:21.800
all the way, anywhere up from negative infinity, up until 12.
00:33:21.800 --> 00:33:25.800
Now, we ask ourselves, "Can we actually achieve 12?" Yes, we can.
00:33:25.800 --> 00:33:33.200
We can actually get to 12, so we include 12; so our range is from negative infinity to 12--there is our answer.
00:33:33.200 --> 00:33:40.500
The final one: we have a word problem: Give the area of a square, A, as a function of the square's perimeter, p.
00:33:40.500 --> 00:33:45.600
And then, also say what is the domain of the area as a function of the perimeter.
00:33:45.600 --> 00:33:49.900
First, as we talked about in the word problems, let's set up what our variables are.
00:33:49.900 --> 00:34:10.800
Nicely, this problem already gave us our variables; but we will just remind ourselves: A is the area of the square, and p is the perimeter of the square.
00:34:10.800 --> 00:34:15.900
So, it also probably wouldn't hurt to draw a picture, so we could see what is going on a little more easily.
00:34:15.900 --> 00:34:25.000
We have a square here; here is our square, and we are talking about the area of it and the perimeter of it.
00:34:25.000 --> 00:34:29.500
So, that is everything that we don't immediately know: we don't know the area; we don't know the perimeter.
00:34:29.500 --> 00:34:33.500
They are going to be somehow connected, because we somehow want to be able to make a function out of area,
00:34:33.500 --> 00:34:36.600
where we plug in the perimeter, and it gives out an area.
00:34:36.600 --> 00:34:41.300
We basically want an equation that has area on the left, and then things involving perimeter.
00:34:41.300 --> 00:34:46.600
We are solving for area in terms of perimeter; that is another way of looking at what this function is going to be.
00:34:46.600 --> 00:34:52.600
We need some way to be able to connect these two ideas: how can we connect the area of a square to its perimeter?
00:34:52.600 --> 00:34:58.900
Well, maybe we don't see a way right away; but let's just think, "Well, how do you find the area of a square?"
00:34:58.900 --> 00:35:05.400
Well, it is its side times its side, its side squared; so the area of a square...
00:35:05.400 --> 00:35:14.000
Now, we might as well go back, and we will set up a new variable--we didn't have that before--side of square.
00:35:14.000 --> 00:35:20.000
A side of the square is a way to get our area; so area equals side squared.
00:35:20.000 --> 00:35:23.000
Now, we still want some way to connect the area to the perimeter.
00:35:23.000 --> 00:35:28.900
So, what we want is...well, we might not be able to correct them directly, but we have area connected to sides.
00:35:28.900 --> 00:35:35.100
Maybe we can connect perimeter to side...oh, right, yes...if you have forgotten what a perimeter is, what do you do?
00:35:35.100 --> 00:35:40.000
You just go and look it up: you have access to all sorts of information at your fingertips--it is so easy.
00:35:40.000 --> 00:35:43.900
If you look up perimeter, thinking, "Oh, I have heard this before; I can't remember what it is,"
00:35:43.900 --> 00:35:48.100
type it into an Internet search; the next thing you know, you will have a definition for what perimeter is.
00:35:48.100 --> 00:36:00.000
So, perimeter is all of the sides added together; we have four sides, so perimeter is equal to side + side + side + side--
00:36:00.000 --> 00:36:04.400
all of the sides of our square, or 4s.
00:36:04.400 --> 00:36:08.300
So now, we have a way of being able to have area talk to perimeter.
00:36:08.300 --> 00:36:18.200
Area equals side squared; perimeter equals 4 times side; so perimeter/4 equals side.
00:36:18.200 --> 00:36:22.700
Now, we can take this, and we can plug it in here.
00:36:22.700 --> 00:36:30.000
Area equals...since we are plugging and we are substituting, we do it with parentheses...squared.
00:36:30.000 --> 00:36:36.600
Area equals perimeter squared, over 16 (we have to square the top and the bottom).
00:36:36.600 --> 00:36:41.300
And there we are--we can think of area equals perimeter squared over 16 as a function,
00:36:41.300 --> 00:36:45.000
because it only depends on what we plug in for perimeter.
00:36:45.000 --> 00:36:50.200
Area will vary as we put in different things; so we can think of it as a function acting on perimeter.
00:36:50.200 --> 00:36:54.100
We plug in the number from perimeter, and it gives out what the area has to be.
00:36:54.100 --> 00:37:03.200
So, we can just rewrite this as: area is a function of perimeter, or it is equal to the perimeter squared, over 16.
00:37:03.200 --> 00:37:08.900
It is just a different way of thinking about it: we can think of it as an equation, or we can think of it as a function
00:37:08.900 --> 00:37:12.700
where it just works the exact same way that the equation worked.
00:37:12.700 --> 00:37:18.000
There is no functional difference between area of perimeter equals perimeter squared over 16--
00:37:18.000 --> 00:37:25.000
the area based on a function using our perimeter equals p²/16--compared to area equals p²/16.
00:37:25.000 --> 00:37:29.300
They have the same effect; it is just two slightly different ways of talking about it.
00:37:29.300 --> 00:37:34.900
But in either case, it is plugging in a number for perimeter, then figuring out what the area has to be.
00:37:34.900 --> 00:37:39.900
That is a function for area as a function of a square's perimeter.
00:37:39.900 --> 00:37:49.100
Now, how can we get its domain? We talked about, before: the domain is everything that we can plug in without breaking it.
00:37:49.100 --> 00:37:52.400
Now, that is mostly true; but there is one little thing here.
00:37:52.400 --> 00:37:58.500
The domain also has to make sense; we can't break the world.
00:37:58.500 --> 00:38:03.900
We wouldn't break this function...we could plug anything we want into this function.
00:38:03.900 --> 00:38:08.900
You plug in any real number in for p, and it would make sense; we would get a number out of it.
00:38:08.900 --> 00:38:14.300
You can plug in 50; you can plug in 0; you could plug in -10; it makes sense--we would get a number out of it.
00:38:14.300 --> 00:38:27.300
But the domain has to make sense; it doesn't have to just make sense in our function--it has to make sense in how we have thought about the function.
00:38:27.300 --> 00:38:31.500
How did we think about the function? It is a square, right?
00:38:31.500 --> 00:38:36.900
It is a real object--it is a thing; we could talk about its shape and how its dimensions are.
00:38:36.900 --> 00:38:44.400
Would it make sense for it to have a perimeter of a negative? No, because it doesn't make sense for the sides to be negative.
00:38:44.400 --> 00:38:49.300
Would it make sense for the perimeter to be 0? No, because then it would just be a speck--it wouldn't be a square.
00:38:49.300 --> 00:38:54.800
There would be no area possible to be contained inside, because we would have no side lengths if we had a perimeter of 0.
00:38:54.800 --> 00:39:03.800
So, it must be the case that our p, perimeter, is allowed to vary only from 0 up to infinity,
00:39:03.800 --> 00:39:07.800
because it can't have a domain below 0, and it can't have a domain of 0,
00:39:07.800 --> 00:39:13.900
because while it doesn't break our function itself, it breaks the idea of what the function means.
00:39:13.900 --> 00:39:19.000
It is meaningless to talk about plugging in a perimeter that is negative or a perimeter that is 0,
00:39:19.000 --> 00:39:22.000
because then it is not the perimeter of anything; we don't actually have a shape there.
00:39:22.000 --> 00:39:25.300
We have to be having our domain make sense, as well.
00:39:25.300 --> 00:39:27.700
If we just have a function, it can't break the function.
00:39:27.700 --> 00:39:33.900
But if we have the function in the context of a word problem, it also has to make sense with everything else happening in the word problem.
00:39:33.900 --> 00:39:37.900
All right, I hope that all made sense, because that just laid an important groundwork.
00:39:37.900 --> 00:39:41.200
You are going to need to know this for the rest of your time in math.
00:39:41.200 --> 00:39:42.600
So, it is really great that we got this covered here.
00:39:42.600 --> 00:39:46.300
Having a really strong understanding of what it means for something to be a function
00:39:46.300 --> 00:39:48.300
is going to help you out in so many different places in math.
00:39:48.300 --> 00:39:51.600
It is going to help you with all sorts of things--it is really great that we covered that here.
00:39:51.600 --> 00:39:54.000
All right, see you at Educator.com later--goodbye!