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Let's

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get going with more examples of function problems, and

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hopefully as we keep doing this, you're going to get

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the idea of how all this stuff works.

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So let's do another problem.

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I'll use green this time.

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Let me clear everything.

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So I'll show you-- I showed you that 1, you could define a

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function as just kind of a standard algebraic expression,

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you could also do it a kind of if number is odd, this is what

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you do, if a number is this, is what you do.

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You could also define a function visually.

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Let's say-- let me draw a graph, and I'll use the line

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tool so it's a reasonably neat graph-- that's

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the x-axis there.

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That's pretty good.

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And let's draw the f of x-axis, or you might be used to calling

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that the y-axis, but-- OK.

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I almost had it vertical, but let's see.

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Let's draw a few slashes here.

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And a couple here, like this.

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Sorry if you're getting bored while I draw this graph.

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I should really have some type of tool so that the

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graphs just show up.

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Let me draw a-- let's say that-- let me

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draw this function.

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So this is what?

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This is 1, 2, 3, 4, 5, this is negative 5, this is 5, this

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is 5, this is negative 5.

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And this is x-axis, and this is-- we'll call

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this the f of x-axis.

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Now that might not seem obvious to you at first, but all this

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is saying is let's say when x is equal to negative 5, this

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function-- I'm creating a function definition-- let's say

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it equals 2, that's negative 1, that stays the same, that stays

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the same, then it goes to here, and then it goes to here, to

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here, and then-- let's see.

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I hope I'm not boring you.

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And it just keeps moving up.

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Let me see, what would this look like-- this

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would look like this.

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So if I-- you might think I'm doing something very strange

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right now, but just bear with me while I draw this.

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I hope I don't mess up too much.

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And, see, one like that.

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See one like that.

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So we're like, Sal, this is a very strange looking graph.

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And it is.

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But what this is, is this is a function definition.

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This tells you whenever I input an x, at least for the x's that

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we can see on the graph, this graph tell me what

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f of x equals.

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So if x is equal to negative 5, f of x would equal plus 2.

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And we could draw a couple of examples.

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f of 0, well we go to 0 on the x-axis, and we say

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f of 0 is equal to 0.

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f of 1 is equal to-- well, we go to x equal to 1, and we

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just see where the chart is, well, it equals negative 1.

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I think you get the idea.

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This isn't too difficult, but this is a function definition.

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So we've defined this graph right here as f of x.

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So if that graph-- that's the graph of f of x, and let's say

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that we define g of x is equal to f of x-- let's say

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it's equal to f of x squared minus f of x.

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And let's say that h of x is equal to 3 minus x.

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So what if I were to ask you, what is h of g of negative 1?

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So just like we did in the previous problems, first we'll

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say, well, let's try to figure out what g of negative 1 is,

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and then we can substitute that into h of x.

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So g of negative 1 is equal to-- and this is how I do it.

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There's no trick to it.

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Wherever you see the x, you just substitute it with the

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number that you're saying is now the value for x.

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So you say, well, that's equal to f of negative 1 squared

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minus f of negative 1.

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All I did is at g of negative 1, I just substituted

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it wherever I saw an x.

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Well what's f of negative 1?

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Well, when x is equal to negative 1, f of

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x is equal to 1.

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So f of negative 1-- let's write that, f of negative

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1 is equal to 1.

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So g of negative 1 is equal to-- well, that's just

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1 squared minus 1, well that equals 0.

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Because f of negative 1 is 1, so it's 1 squared minus

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1 that equals 1 minus 1.

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0.

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So g of negative 1 is 0, so this is the

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same thing as h of 0.

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Because g of negative 1, we just figured out is 0.

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h of 0, we just take that 0 and substitute it here, so it's 3

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minus 0, so that just equals 3.

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And we solved the problem.

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Let's do another example, and I don't want to erase my graph

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since I took four minutes to actually draw it, let me

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erase what we just did here.

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And what you might want to do after you watch it the first

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time-- and this isn't true just of this video, actually of all

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the videos-- but especially the functions, after watching it

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once, you might want to rewatch it and pause it right after I

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give you the problem and try to do it yourself, and then see--

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and if you get stuck, you can play it, or if you get an

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answer, just you can play the video and make sure that

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we did the same way.

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Let's see.

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I'm going to create another definition

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for g of x this time.

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Let's say that g of x-- oh whoops, I was trying to write

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in black-- let's say that g of x is equal to f of x

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squared plus f of x plus 2.

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So now, in this case, what is g of-- let's pick a random

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number-- what is g of minus-- no, let's pick a, let's

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say-- what is g of minus 2?

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After we try and pick a number that we could find

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an actual solution for.

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Well g of minus 2, wherever we see the x, x is not

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going to be minus 2.

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That is equal to f of minus 2 squared plus

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f of minus 2 plus 2.

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All we did is wherever we saw an x, we substituted

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it, minus 2 there.

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And let's simplify that.

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Well, f of minus 2 squared, we know what minus 2 squared is,

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that's the same thing as f of 4, plus f of minus 2 plus 2.

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That's 0.

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Plus f of 0.

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And now we just figure out what f of 4 and f of 0 is.

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Well, f of 4, we go where x equals r, it's right here,

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and when x equals 4, f of 4 is equal to 2.

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So this is equal to 2 plus f of 0.

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And just as a reminder, this is the definition of f.

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We didn't define it in terms of an algebraic expression, we

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defined in terms of an actual visual graph.

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So what's f of 0? f of 0 is 0.

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When x is equal to 0-- f of 0 is 0 so that's 2 plus 0-- so g

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of negative 2 is equal to 2.

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An interesting thing, you might want to make problems like this

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for yourself and keep experimenting with different

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types of functions, and a very interesting thing would

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actually be to graph g of x, and actually that's a

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good idea, I think.

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I think maybe we'll do that in the future modules to kind of

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play with functions and actually to try graph

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the functions and see how they turn out.

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I will-- I don't know if I have enough time-- actually, I'm

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going to wait until the next lecture to do a couple

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more examples.

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I want to do as many examples on the functions as I can with

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you, because I think as you keep watching and watching the

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function problems and seeing more and more variations on

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functions, you'll see both how general of a concept this is,

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and hopefully you'll get an idea of how the functions

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actually work.

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Well, I'll see you in the next lecture.

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Have fun.

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