Calculus: Functions

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The Basics
Precalculus Review
Functions Page [1 of 1]
Okay, well, now, what was this, what does that warm-up question have to do with this previous issue we were talking about, about finding instantaneous rate of change? Well, the answer is that we first have to adopt a little bit of language to be able to tackle that problem. Remember, we tried the zero over zero thing, which, as I mentioned, avoid. So, we need to develop a little bit of language so we can speak the, the words of mathematics. And so, and so what we'd do first is actually go on location right now, we're going to leave the studio, and I'm going to actually show you, try to inspire what we are about to think about. So, join me right now, okay?
Well, here we are on location at the ATM machine, like there are places and the really neat thing about the ATM machines is that all you have to do is put in your account number, and you can actually find out how much money you have in your account. And the neat thing is all it has is my account number. It goes off, it's like a little machine that goes off and looks something up and spits out the one unique answer, the one unique answer of how much money I have in my account. Let's see what it can spit out. So, here I have exactly how much money is in my account, or how much money there should be in my account. Anyway, the point is that just putting in as input, my account number, I can find out exactly how much money I have in my account. Where does this lead us mathematically? Let's go back to the studio and find out. I'll see you there. Bye.
Okay, well, I hope you enjoyed the little ATM adventure, and the point of that, of course, was that the ATM machine is sort of interesting in that it takes in your account number and it spits out a unique number, and that number represents how much money you have, hopefully, in your, in your account. Now that actually is an example of a math idea, which is called a function. Now, I know you've, you've heard of a function before. But I just wanted to remind you of this because that really is the language of this course. So this is sort of now like a little mini French, or German, or Russian kind of a vignette here, because we can now learn the language of mathematics.
The language of mathematics is function. Now, so, what's a function? Well, basically, the idea of a function is, just like we saw at the ATM. We put a piece of information in and the function is like a little machine that takes it, massages it, generates it, throws it around, opey, doopey, doopey do, but at the end of the day, patink! One thing gets spit out and that one thing is, basically, the result of the function.
So, you can think of the function like this. You can think of the function as a sort of mysterious machine, you know, remember, we are on the web, folks, so, you know, we got to sort of talk about technology as much as we can. So, so here is the function, and I'll call it F for function. I'm just calling it F for function, it's the machine, the machine is this. And you put something in, so you put something in, like let me put in something, I'll call it just X for lack of a more creative name. You put that into the machine and it rattles around in there, and ding, dada ding, dada ding, dada ding and what happens at the end of the day, bing, out comes the answer. And the answer I'm actually going to write as F of X. So, now what I'm going to do is actually show you some notation. But the idea of a function is just - it's something where you put one thing in and then one thing comes out.
Now let me show you how we actually write that. One way of writing it, by the way, would just be, let me give you an example of a function, might be Y equals X squared. That's an example of a function and here's how it works. Basically, you put X; you put the value that you want in for X, okay? So, you put in some number like seven, and then the little machine is actually this machine here. It's the square machine. It, it takes a number and just squares it and it spits out, what we're going to call Y in this, for this moment, and, so, if you put in a five, for example, ding dada ding, it spits out twenty five. Okay, not too hard. But now we're not going to actually use this notation. What I would like us to use is this notation here. So, I've got to actually write this, again, not an idea, it's just some language, but it more captures, I think it better captures the spirit of this little machine. Because what this says is F is a function and it has to know this input. It has to know what X is, and then once you give me X, this function goes off, and, you know what it does? It produces X squared. So, for example, in that previous example, here on this side I have to write up the following: Find what Y equals when X equals five. Wheew, that was long. Here, all I have to write is find F of five, because what does F of five mean? It means find out what the machine F will do if you input five, and then you instantly pop back, well, it would be five squared, which equals twenty five - the language of functions.
Let's try another example. So, to illustrate this a little bit further, let me try to actually capture the spirit of what's going on, this notion of the machine here. So, in fact, what's going on here, let me actually write down a, a problem, let me write a function here, so could write down F of X equals, how about three X squared, I'm just making this up on the fly, minus two X plus thirty five. This is a much more exotic function than the previous one because if you want to know where X gets sort of mapped to, or gets generated to, or it's pushed out to, you actually compute this long thing, you got to take that number and square it, then multiply it by three, then subtract from it two times that number and then add thirty five. Wheew!
But let me try to illustrate this with an example, in fact, a way of thinking about this machine. I'll actually show you one. You see, I'm going to show you the machine. You're thinking is there a machine. Yes, there is the machine, and I've got one right for you here. I've got one. Here's the machine. This machine, by the way, is this machine right here. So, if you put something in what's going to be spit out, in fact, is this. Let me try to illustrate that with an example. So, if we input, for example, let's say zero, now let's now try to figure out what F of zero would be. Okay, well, what I would have to do is wherever I see an X, all occurrences of X here, what we have to do is we have to plug in zero and see what we get. Let's try that right now. So, here is zero, and I'm going to actually just put it right into the machine here, and if I put it into the machine and turn the machine. Look what happens, out pops thirty-five. Is that right? Well, let's see. If we plug in zero everywhere for X, then that's going to be zero, that's going to be zero, and this would, indeed be thirty-five. So, in fact, that's the right answer. See how it works. I put in zero and out came thirty-five.
Let's try another one. Isn't this, isn't this great? I really like this; I really like this machine, by the way. I personally like it a lot. Let's try another one. Let's try F and see where two goes. By the way, let me tell you how, how you say this in words. The way you say this in words is we read this as F of two, F of two. In fact, let me write that down for you right here. F, this is, sort of, for example, suppose you're reading a kids book to a niece, a nephew, or, or a younger sibling, okay, and it says this. How do you read that in a kid's book? You would say F of two, or, in general, F of X, that's how you read it. But you know what it means. It means just plug in a two wherever you see an X. Now what would that be? Well, let's see, I've got a two here. Should we try it? Let's try it, let's try to put it all in and just see if we can make progress here. Well, there it goes, it's going into the machine. It's coming out of the machine, and let's see what we got. We've got forty-one.
Well, let's see if that's right. Well, this is actually going to take us a, a couple of seconds here to think about, if that's the right answer. Wherever I see an X, I have to put in a two. So here I could put in a two, that's two squared, so that's four, and then I take the four and multiply it by three and that gives me twelve. So this is going to be a twelve, but then I have to subtract off two times two, which is a four, and then I have to add thirty-five. So what does that equal? Well, this seems to give me forty-seven, minus the four, and, happily, that is not the answer. (Laugh). Well, now this is good, this is good, because the answer seems to be forty-three. Okay, so what went wrong there? Well, actually, this was one of those little things in the web, you know when the things in the web, and things get sort of messed up and all of a sudden like you, you freeze, you know, freeze, freeze. Well, that's exactly what happened here. And I know you saying, "No, that can't be," but it actually is, and I'll prove it to you right now, because here is the, the actual thing that came out. You see that? Look at that. So, we were absolutely right again. This always seems to work without fail. (Laugh). Yeah, okay, all right.
Now, as, as one final example, let me just do one last example, I'm going to save myself here, because it's not looking good. Okay, well we'll just do it right now, because let's actually see what would F of five dollars be? Well, how would we find that out? Wherever we see an X, we're going to plug in five dollars. Let's try that right now. If we put in five dollars for X here, then we see five dollars squared, that's twenty-five dollars, multiplied by three, would be seventy-five dollars. Then I subtract off two times five dollars, and two times five dollars is ten dollars, so I have to subtract ten dollars and then add thirty-five dollars. So that's a net gain of plus twenty-five dollars, and that gives me one hundred dollars. Now, you probably are not sure if that's really right or not, so let's try it. Here's the five-dollar bill, and let's see what happens. We'll put that in here, I hope we're not going to have a little, same as we did it the last time, let's see what happens. Look at that, folks! Right in front of your eyes, look at that. Takes a little while, but these are big numbers now. Look at that. Can you see that? Ooh, can you smell that, that is the smell of a brand new crisp hundred-dollar bill. Folks, if nothing else, this shows math pays, math pays. This one I'm keeping, isn't that great. Anyway, so there's the, there's the idea of a function.
Let's try a couple more basic examples here really fast; same function, but now let's just try some other, other variations on the theme. For example, suppose that I plug in something else for X. What could I plug in? The point is I could plug in anything I want. For example, bear with me here, what is F of, and here I have a quarter. I don't know if you can see that or not, but that's a quarter, it really is, really is a quarter. What is F of a quarter? What's the recipe? By the way, at this point you might be saying, "Gosh, if he does this one more time I'm going to get sick." Stay with me, this is really an important point, and if you got it and say there's no problem to this, well, great. But it's important to really capture this now, no matter what I put in here, what do you do? Wherever you see an X, you're going to replace it, in this case, by a quarter. So, here you see three times quarter squared, minus two times quarter, plus thirty-five. Not bad. You can even try more exotic examples. How about F of chicken, ha, ha, you can put in poultry, you can put in whatever you want. If you're a vegetarian, you can put in a carrot. It's okay. What's the answer? The answer is it's three times chicken squared, minus two times chicken, plus thirty-five. Always the same. The point is that this X, which maybe, is intimidating to some, is just a placeholder. So, we could put in here, you know, F of blank, fill in the blank, it equals three times fill in the blank squared, minus two times fill in the blank, plus thirty five. Whatever you want to put in there is fine, whether it's numbers, whether it's money, whether it's quarters, whether it's chickens. No matter what, you just follow this recipe. In fact, let me just show you and then indicate that you can do this kind of thing with even more abstract things, I mean, something even more abstract than the chicken. I know what you're thinking, saying, "Gee, the chicken sounds pretty abstract to me." Ha, ha, ha, ha. I'm getting hungry, by the way, are you? I'm getting hungry. There's no chickens, no live chickens here today, folks, I'm sorry about that.
Okay, how about this though? How about F of A, in fact, right now I'm going to stop and I'm going to give you a chance to write down exactly what is F of A. Go ahead. Okay, well, what is F of A? The rule is a very simple rule. Wherever I see an X, I plug in A. So, this would be three times A squared, minus two A, plus thirty-five. No problem.
Okay. I want to try one last example with you here on this one, and this one is called the most abstract and the trickiest of all. It turns out also to be the most valuable and the most useful for what we're going to do. What if I put in F of something like A plus B? What do you do now? In fact, why don't you try right now and see if you can figure out what the answer is for F of A plus B and, wait, wait, wait, before you stop, and before I stop, and before you try, try to simplify it as much as you can. Okay, try to see if you can, sort of, figure it all out and write it in the simplest form that you can. Okay. Shoot.
All right. Well, let's try it together and see what happens. The recipe is always to write down wherever I see an X, this. Now, I'm going to do this problem right now wrong. Okay? So, in fact, let me actually indicate that by drawing a line right here to tell you that what I'm about to do is wrong, but this is a great guess, in fact, this is sort of a great, great guess a lot students do. They would say okay, wherever I see an X, I'm going to put in that whole thing, so they say, okay, that's going to be three A plus B squared, minus two A plus B, plus thirty five. Now maybe that actually looks right to you and if it does, by the way, that's great, because, like I'm telling you, this is an old classic. But I'm telling you right now, this is wrong. In fact, I'll give you a second right now. See if you can figure out why this is wrong, especially if it's smells so correct to you and you feel that it's right, `How could he be saying it's wrong, when I feel so right about this." Well, think about it for a second and see if you can, if you can figure it out, and I'll tell you.
Okay, well, maybe you realize that this is actually pretty close to being correct, but the problem is that it's all of X that has to be squared, and in this case X is A plus B. So, all of that stuff, all of the A plus B, has to be squared, and what I wrote here was not that. I wrote A plus B squared. So, if you want to really show me that it's the whole thing that's being squared, I've got to put parenthesis around that, you see, I've got to put parenthesis all around that, and then square it. Similarly here, here I've just been taking, I'm supposed to take negative two times all of that. Well, I looked at negative two times the A and I forgot about the B, the B is all by itself. Oh, it's so lonely; it desperately wants to be multiplied by negative two. How would you do that? You'd have to put parenthesis around that and then subtract, multiplied by the negative two. Just as we've seen before, the idea is, it's a placeholder. The X is a placeholder and you got to plug in. So, the actual retail value, oh, that's another program. The actual answer here, would be what? It would be three times the quantity A plus B, all squared, minus two times the quantity A plus B, plus thirty-five. You see the parenthesis actually really grab those things together.
Now, in fact, I did ask you, there's a little extra bonus problem in this, and that's, the bonus question was, can you simplify that? I just want to remind you how to simplify things just a little bit here, in case you're a little rusty. How would you foil that out? A great, great mistake, by the way, in fact, let me lower this back down, just to write this where I can get rid of it really fast. A great mistake is that if you take A plus B - this is an old favorite, by the way, that, and square it, that equals A squared plus B squared. Remember, remember that? Yeah, that's a, that's a great one, that's a great classic, and I want you to look at that. I want you to enjoy that right now. In fact, I'm going to make it all yellow, because, I mean, this is a really, in fact, you know what? I'm going to even put it in a little fancier color. Look at that. I want you to look at this. I want you to enjoy it, take it in. It's an interesting vista and I want you then to never, ever do this again. The answer is that you have to actually foil all this out. So, let me remind you how the foil goes really fast here. The foil, what would this equal? Well, this would be three times, and if I foiled all this out, A plus B, times A plus B, I would get A squared plus two AB plus B squared, and then I subtract off, and all that should distribute that, that, that two, so I subtract off two A plus two B plus thirty-five. And if you simplify that a little bit more and distribute, you'd see three A squared plus three times two, is six AB, plus three B squared, minus two A, and notice I'm going to subtract that, the minus sign gets hit to the two B, plus thirty-five. And that long looking answer is actually the answer here.
Okay, so, so that is at the heart of what functions are all about. And one last note, by the way, don't get all hung up and don't marry F of X, because I could have written a lot of other symbols for this thing. I could have written, for example, uh, I don't know, G of T equals T squared. Same thing. Wherever you put it, whatever you put in here for T, that's what you've got to do for G, G is defined to be that thing squared. So, there's just a lot of symbols, I could put any symbol I want here. Just remember, it's a function and whatever I plug in here, this function goes off, this machine's name is G. It goes off, da, da, da, da, da, and does that. This is just the placeholder and that's just the name of the function. So, that's sort of the spirit of functions, and, believe it or not, that's going to give us the language in which we can talk about more interesting issues of mathematics.
Okay, so why don't you try some of these questions on your own, right now, and when we come back we're going to try to do this thing about the zero over zero and, and how we get closer and closer to that. That, right now, is a cliffhanger, folks. But, we're going to get to it, right in a second. Okay, see you in a bit.