# Introduction to functionsÂ (old)

## Video transcript

Welcome to the presentation
on functions. Functions are something that,
when I first learned it, it was kind of like I had a
combination of I was 1, confused, and at the same time,
I was like, well what's even the point of learning this? So hopefully, at least in this
introduction lecture, we can get at least a very general
sense of what a function is and why it might be useful. So let's just start off
with just the overall concept of a function. A function is something that
you can give it an input-- and we'll start with just one
input, but actually you can give it multiple inputs-- you
give a function an input, let's call that input x. And you can view a function
as-- I guess a bunch of different ways you can view it. I don't know if you're
familiar with the concept of a black box. A black box is kind of a box,
you don't know what's inside of it, but if you put something
into it like this x, and let's call that box-- let's say the
function is called f, then it'll output what
we call f of x. I know this terminology might
seem a little confusing at first, but let's make some-- I
guess, let's define what's inside the box in
different ways. Let's say that the function
was-- let's say that f of x is equal to x squared plus 1. Then, if I were to say
what is f of-- let's say, what's f of 2? Well that means we're taking
2 and we're going to put it into the box. And I want to know what
comes out of the box when I put 2 into it. Well inside the box, we know
we do this to the input. We take the x, we square it,
and we add 1, so f of 2 is 2 squared, which is 4, plus 1. Which is equal to 5. I know what you're thinking. Probably like, well, Sal, this
just seems like a very convoluted way of substituting
x into an equation and just finding out the result. And I agree with you right now. But as you'll see, a function
can become kind of a more general thing than
just an equation. For example, let me say-- let
me actually-- actually not, let me not erase this. Let me define a
function as this. f of x is equal to x squared
plus 1, if x is even, and it equals x squared
minus 1 if x is odd. I know this would have been--
this is something that we've never really seen before. This isn't just what I would
call an analytic expression, this isn't just x plus
something squared. We're actually saying,
depending on what type of x you put in, we're going to do a
different thing to that x. So let me ask you a question. What's f of 2 in this example? Well if we put 2 here, it says
if x is even you do this one, if x is odd you do this one. Well, 2 is even, so
we do this top one. So we'd say 2 squared plus
1, well that equals 5. But then, what's f of 3? Well if we put the 3 in
here, we'd use this case, because 3 is odd. So we do 3 squared minus
1. f of 3 is equal to 8. So notice, this was a little
bit more I guess you could even say abstract or
unusual in this case. I'm going to keep doing
examples of functions and I'm going to show you how
general this idea can be. And if you get confused, I'm
going to show you that the actual function problems
you're going to encounter are actually not that hard to do. I just want to make sure that
you least get exposed to kind of the general idea of
what a function is. You can view almost anything
in the world as a function. Let's say that there is a
function called Sal, because, you know, that's my name. And I'm a function. Let's say that if you
were to-- let me think. If you were to give me
food, what do I produce? So what is Sal of food? So if you input food into
Sal, what will Sal produce? Well I won't go into some of
the things that I would produce, but I would
produce videos. I would produce math videos
if you gave me food. Math videos. I'm just a function. You give me food and-- and
maybe, actually, maybe I have multiple inputs. Maybe if you give me a food
and a computer, and I would produce math videos for you. And maybe you are a function. I don't know your name. I would like to, but I
don't know your name. And let's say if I were to
input math videos into you, then you will produce-- let's
see, what would you produce? If I gave you math videos, you
would produce A's on tests. A's on your math test. Hopefully you're not taking
someone else's math test. So it's interesting. If you give-- well, let's
take the computer away. Let's say that all
Sal needs is food. Which is kind of true. So if you put food into
Sal, Sal of food, he produces math videos. And if I were to put math
videos into you, then you produce A's on your math test. So let's think of an
interesting problem. What is you of Sal of food? I know this seems very
ridiculous, but I actually think we might be going
someplace, so we might be getting somewhere with
this kind of idea. Well, first we would try to
figure out what is Sal of food. Well, we already figured out if
you put food into Sal, Sal of food is equal to math videos. So this is the same thing as
you of-- I'm trying to confuse you-- you of math videos. And I already determined, we
already said, well, if you put math videos into the function
called you, whatever your name might be, then it produces
A's on your math test. So that you of math videos
equals A's on your math test. So you of Sal of food will
produce A's on your math test. And notice, we just said
what happens when you put food into Sal. This could-- would be a very
different outcome if you put, like, if you replaced food
with let's say poison. Because if you put poison into
Sal, Sal of poison-- not that I would recommend that you did
this-- Sal of poison would equal death. No, no, I shouldn't say
something, so no no no no. Well you get the idea. There wouldn't be math videos. Anyway. Let me move on. So with that kind of-- I'm not
so clear whether that would be a useful example with the
food and the math videos. Let's do some actual
problems using functions. So if I were to tell you that I
had one function, called f of x is equal to x plus 2, and I had
another function that said g of x is equal to x
squared minus 1. If I were to ask you
what g of f of 3 is. Well the first thing we want to
do is evaluate what f of 3 is. So if you-- the 3 would replace
the x, so f of 3 is equal to 3 plus 2, which equals 5. So g of f of 3 is the same
thing as g of 5, because f of three is equal to 5. Sorry for the little
bit of messiness. So then, what's g of 5? Well, then we take this 5, and
we put it in in place of this x, so g of 5 is 5 squared, 25,
minus 1, which equals 24. So g of f of 3 is equal to 24. Hopefully that gives you a
taste of what a function is all about, and I really apologize
if I have either confused or scared you with the Sal
food/poison math video example. But in the next set of
presentations, I'm going to do a lot more of these examples,
and I think you'll get the idea of at least how to do these
problems that you might see on your math tests, and maybe get
a sense of what functions are all about. See you in the next video. Bye.