Introduction to functions An introduction to functions.
Introduction to functions
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- 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 at least get exposed
- to 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 that's interesting. If you give...
- 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?
- 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 we put food into Sal, you know.
- 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.
- 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 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.
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