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Welcome to the presentation on functions.

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Functions are something that, when I first learned it, it

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was kind of like I had a combination of I was 1,

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confused, and at the same time, I was like, well what's even

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the point of learning this?

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So hopefully, at least in this introduction lecture, we can

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get at least a very general sense of what a function is

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and why it might be useful.

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So let's just start off with just the overall

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concept of a function.

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A function is something that you can give it an input-- and

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we'll start with just one input, but actually you can

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give it multiple inputs-- you give a function an input,

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let's call that input x.

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And you can view a function as-- I guess a bunch of

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different ways you can view it.

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I don't know if you're familiar with the

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concept of a black box.

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A black box is kind of a box, you don't know what's inside of

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it, but if you put something into it like this x, and let's

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call that box-- let's say the function is called f, then

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it'll output what we call f of x.

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I know this terminology might seem a little confusing at

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first, but let's make some-- I guess, let's define what's

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inside the box in different ways.

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Let's say that the function was-- let's say that f of x is

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equal to x squared plus 1.

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Then, if I were to say what is f of-- let's

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say, what's f of 2?

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Well that means we're taking 2 and we're going to

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put it into the box.

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And I want to know what comes out of the box

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when I put 2 into it.

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Well inside the box, we know we do this to the input.

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We take the x, we square it, and we add 1, so f of 2 is 2

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squared, which is 4, plus 1.

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Which is equal to 5.

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I know what you're thinking.

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Probably like, well, Sal, this just seems like a very

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convoluted way of substituting x into an equation and just

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finding out the result.

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And I agree with you right now.

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But as you'll see, a function can become kind of a more

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general thing than just an equation.

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For example, let me say-- let me actually-- actually not,

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let me not erase this.

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Let me define a function as this.

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f of x is equal to x squared plus 1, if x is even,

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and it equals x squared minus 1 if x is odd.

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I know this would have been-- this is something that we've

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never really seen before.

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This isn't just what I would call an analytic expression,

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this isn't just x plus something squared.

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We're actually saying, depending on what type of x you

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put in, we're going to do a different thing to that x.

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So let me ask you a question.

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What's f of 2 in this example?

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Well if we put 2 here, it says if x is even you do this one,

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if x is odd you do this one.

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Well, 2 is even, so we do this top one.

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So we'd say 2 squared plus 1, well that equals 5.

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But then, what's f of 3?

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Well if we put the 3 in here, we'd use this

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case, because 3 is odd.

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So we do 3 squared minus 1. f of 3 is equal to 8.

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So notice, this was a little bit more I guess you could

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even say abstract or unusual in this case.

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I'm going to keep doing examples of functions and

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I'm going to show you how general this idea can be.

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And if you get confused, I'm going to show you that the

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actual function problems you're going to encounter are

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actually not that hard to do.

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I just want to make sure that you least get exposed to kind

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of the general idea of what a function is.

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You can view almost anything in the world as a function.

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Let's say that there is a function called Sal, because,

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you know, that's my name.

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And I'm a function.

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Let's say that if you were to-- let me think.

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If you were to give me food, what do I produce?

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So what is Sal of food?

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So if you input food into Sal, what will Sal produce?

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Well I won't go into some of the things that I would

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produce, but I would produce videos.

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I would produce math videos if you gave me food.

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Math videos.

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I'm just a function.

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You give me food and-- and maybe, actually, maybe

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I have multiple inputs.

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Maybe if you give me a food and a computer, and I would

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produce math videos for you.

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And maybe you are a function.

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I don't know your name.

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I would like to, but I don't know your name.

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And let's say if I were to input math videos into you,

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then you will produce-- let's see, what would you produce?

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If I gave you math videos, you would produce A's on tests.

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A's on your math test.

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Hopefully you're not taking someone else's math test.

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So it's interesting.

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If you give-- well, let's take the computer away.

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Let's say that all Sal needs is food.

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Which is kind of true.

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So if you put food into Sal, Sal of food, he

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produces math videos.

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And if I were to put math videos into you, then you

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produce A's on your math test.

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So let's think of an interesting problem.

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What is you of Sal of food?

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I know this seems very ridiculous, but I actually

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think we might be going someplace, so we might be

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getting somewhere with this kind of idea.

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Well, first we would try to figure out what is Sal of food.

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Well, we already figured out if you put food into Sal, Sal of

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food is equal to math videos.

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So this is the same thing as you of-- I'm trying to confuse

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you-- you of math videos.

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And I already determined, we already said, well, if you put

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math videos into the function called you, whatever your name

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might be, then it produces A's on your math test.

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So that you of math videos equals A's on your math test.

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So you of Sal of food will produce A's on your math test.

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And notice, we just said what happens when you

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put food into Sal.

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This could-- would be a very different outcome if you put,

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like, if you replaced food with let's say poison.

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Because if you put poison into Sal, Sal of poison-- not that I

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would recommend that you did this-- Sal of poison

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would equal death.

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No, no, I shouldn't say something, so no no no no.

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Well you get the idea.

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There wouldn't be math videos.

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Anyway.

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Let me move on.

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So with that kind of-- I'm not so clear whether that would be

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a useful example with the food and the math videos.

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Let's do some actual problems using functions.

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So if I were to tell you that I had one function, called f of x

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is equal to x plus 2, and I had another function that said g

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of x is equal to x squared minus 1.

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If I were to ask you what g of f of 3 is.

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Well the first thing we want to do is evaluate what f of 3 is.

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So if you-- the 3 would replace the x, so f of 3 is equal to

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3 plus 2, which equals 5.

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So g of f of 3 is the same thing as g of 5, because f

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of three is equal to 5.

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Sorry for the little bit of messiness.

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So then, what's g of 5?

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Well, then we take this 5, and we put it in in place of this

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x, so g of 5 is 5 squared, 25, minus 1, which equals 24.

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So g of f of 3 is equal to 24.

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Hopefully that gives you a taste of what a function is all

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about, and I really apologize if I have either confused or

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scared you with the Sal food/poison math video example.

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But in the next set of presentations, I'm going to do

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a lot more of these examples, and I think you'll get the idea

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of at least how to do these problems that you might see on

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your math tests, and maybe get a sense of what functions

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are all about.

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See you in the next video.

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Bye.

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