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Welcome to the second presentation on functions.

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So let's take off where we left off before.

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I still apologize -- in retrospect that that

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whole foul food example.

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Well maybe it was helpful, so I'm going to leave it there.

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Let's do some more problems.

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I think the best thing is to keep doing problems with you

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and I think you'll see the example, and hopefully

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you'll actually see that functions are kind of fun.

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Let's do some more problems.

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Let's start off with an example, not too different

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than what we saw before.

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Let's say that g of x is equal to 1 if x is even, and

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it equals 0 if x is odd.

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And let's say f of x is equal to x plus 3 times g of x.

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And let's say -- I'm going to make it really complicated --

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well, actually I'm not going to make it any more

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complicated now.

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So let's try some problems.

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So let's give an example.

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What is f of 5.

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Well, it's really pretty straightforward.

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We take this 5 and we replace it for x in the function f.

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So f of 5 is equal to 5 plus 3 times g of 5, right?

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We just literally took this 5 and replace it everywhere

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where we see an x.

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If instead of a 5, I had like a dog here, it would be f of dog

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would equal dog plus 3 times g of dog.

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Not that that would necessarily make any sense, but

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you get the idea.

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So f of 5 equals 5 plus 3 times g of 5.

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But what does that equal?

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So the 5 stays the same, plus 3 times -- well what's g of 5?

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Well, if we put 5 here, if 5 is even we do 1,

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if five is odd we do 0.

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Well 5 is odd so it's a 0.

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So g of 5 is equal to 0.

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So this is 3 times 0.

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So this equals just 5, right, because 3 times

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0 is equal to 5.

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Well what would be f of 6?

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Well, f of 6 would equal 6 plus 3 times g of 6.

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And once again, that equals 6 plus -- well, this time g of

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6 is, well, 6 is even, so 1.

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So g of 6 is equal to 1.

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So this equals 6 plus 3 times 1.

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So this equals 6 plus 3 which equals 9.

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I think you might be getting the idea now.

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At first when you see a problem with a lot of these functions,

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it seems very confusing.

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But if you just keep taking what's inside of the

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parentheses and replacing that for x and just keep moving

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along that way, you make a lot of progress on these problems.

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Let's try a harder one.

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Let's say I said that f of x is equal to x squared plus 1.

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Let's say that g of x is equal to 2x plus f of x minus 3.

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And h of x is equal to 5x.

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Now I'm going to give you a tough problem.

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What is h of g of x?

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No.

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What is h of g of -- let's pick a number -- let's say 3?

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h of g of 3.

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Actually, we'll do examples in the future where we actually

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could leave the x there and we'll solve for it.

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But let's say this particular example, what is h of g of 3?

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At first you might say wow, this is crazy, Sal, I don't

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know how to even start here.

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But you just take it step-by-step.

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What can we figure out?

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Can we figure out what g of 3 is?

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Well sure.

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We could take the 3 and put it into the function g and

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see what it spits out.

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So let's work on g of 3 first.

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So, g of 3 equals -- well it's 2 times 3, right, we're just

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replacing wherever we see an x with a 3.

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So it's 2 times 3, so that's 6, plus f of -- what, we'll

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just replace the x again.

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3 minus 3, right?

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So this g of 3 is equal to 6 plus f of what?

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3 minus 3 is 0.

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Now we have to figure out f of 0 is.

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We have a definition here for f, so we just figure it out.

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f of 0 is equal to -- well, you replace the 0 here.

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So you get 0 squared, which is 0 plus 1.

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So it's f of 0 is 1.

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So you take that and you replace it for f of 0.

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So you get g of 3 is equal to 6 plus 1.

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So g of 3 is equal to 7, right?

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Now we know what g of 3 is equal to.

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We can substitute that back here.

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So that's the same thing -- we know g of 3 is equal to 7,

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so that's the same thing as h of 7.

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And h of 7 is just equal to 5 times 7 equals 35.

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So I think you're probably a little confused here, and I

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would have been if I was in your shoes.

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But the important thing is when you first see this problem

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you're like what can I tackle first?

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h of g of 3, it seems very confusing.

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Well, g of 3, can I tackle that?

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Sure.

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I have a definition of what the function g does when

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it's given an x, or in this case, was given a 3.

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And that's what we did.

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We figured out what g of 3 was first.

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And g of 3, we just have to do the 3, and we said well that's

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6 plus f of 3 minus 3, right?

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Because we just replaced that x with that 3.

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And we just kept solving.

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We figured out what f of 0 is up here.

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And we got g of 3 equals 7.

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Then we substituted that back in right here.

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We got h of 7 is equal to 35 because it was 5 times 7.

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Let's do some more problems.

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Actually, let's do another example with the same

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set of functions.

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I don't want to keep confusing you with new functions.

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Let me it erase this as fast as I can.

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I think I'm getting faster at this erasing business.

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You can sit and think a little bit about what we just

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did while I erase.

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So let's do another problem.

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What is f of h of 10?

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Well, first we want to figure out what h of 10 is, right?

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Well, we could do it in a different way

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as we'll see later.

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But we can figure out what h of 10 is pretty easily.

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We take the 10, substitute it in for x.

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h of 10 is equal to 5 times x.

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In this case x is 10 so it equals 50.

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So we know h of 10 equals 50.

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So we know h of 10 equals 50, so we substitute

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that back in here.

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So we say f of h of 10 is the same thing as f of 50.

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And then f of 50 is, I think pretty straightforward

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at this point.

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You just take that 50 and replace it back here.

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Well, it's 50 squared plus 1.

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Well, 50 squared is 2,500 plus 1.

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That equals 2,501.

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What is g of h of 1?

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Well, we take h of 1, h of 1 is 5, so this is equal to g of 5.

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And g of 5, we just replace the 5 here, so g of 5 is equal to 2

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times 5 plus f of 5 minus 3.

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We just take wherever we saw an x and replace it with a 5.

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Well, that's equal to 2 times 5 is 10, plus f of 5 minus 3.

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Well 5 minus 3 is 2.

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Plus f of 2.

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What's f of 2?

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Well, 2 squared plus 1 is 5, right? f of 2 is 5

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-- 2 squared plus 1.

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So that equals 10 plus 5 which equals 15.

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If you're still confused, don't worry.

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I'm about to record some more problems that will give you

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even more examples of function problems.

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See you in the next presentation.

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Bye.

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