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Alright, it looks pretty crazy up there is when you're kind of reading it out loud.
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And the combination of functions that's on this is what we're going to be dealing with
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composition.
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And a couple of things I want you to remember when we're looking at functions, remember
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a function as input value and output value.
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And when we're finding the value of a function, we plug in what our input value is to find
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that value.
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Well, composition function is going to work in the same way.
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Now we're going to have two different functions, but what's going to happen is we're actually
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going to have one of our functions served as our input value.
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So let's look at the first one.
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I'm going to say a function f of x.
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So we have our function f of x is going to serve as a function, that means a function
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relationship of another function, g of x.
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So that looks like that.
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So we have g of x, g of x.
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And what we're going to say rather than just saying, you know, our function g of x, what's
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going to our value is going to be f of x, the value we're going to try to find.
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So actually what we're going to look for is g of f of x.
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And that is actually what we're going to be looking at and trying to find.
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For this one, it says, I have a function g of x, which is going to serve as my input
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value, that is going to serve as a function relationship of another value of another function
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f of x.
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So all of you guys can kind of think of that.
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So this one, a function g of x as a function of another function f of x, just sounds just
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doesn't make sense.
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Sounds really crazy.
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So pretty much what I'm really trying to tell you guys to do, actually look how they
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have that.
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So a lot of times what they write is, so a lot of times what we use in this notation
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is the way that I learned it, and this remember it, is we call this golf, just because it's
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missing, it looks like it's missing it out.
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This is like a multiplication symbol, but it's hollow, so it's open, but it's g of x,
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g of x, and then this one is f of g, we call this one fog.
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It's not really an old but it's a little circle, but I don't know, that's the way I
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remember it.
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So that's exactly a lot of times you guys will get this in your book.
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This is the same thing as this.
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So what's happening on this function is I'm plugging in my g of x function into my function
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g of x, and over here what I'm doing is I'm plugging in my function f of x into my function
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g of x.
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So let's look how this is going to work.
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Let's do the first one.
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I have g of x, which is x plus 2.
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So I have x plus 2.
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That's what g of x is.
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g of x equals x plus, actually let's just do it over here.
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x equals x plus 2.
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Well if I want to find g of x, just like what you guys used to evaluate when we evaluate
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it functions at the beginning of the year, when we plugged in a number for that function,
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but I want to find f of 3.
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We plugged in 3 and 4 here x, right?
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Well now we're not plugging in 3, but what we're actually going to do is plug in our
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f of x.
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So f of x is x squared.
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Now we look, is there anything we can simplify with?
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I hear there's no.
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So that is the final answer.
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Now what we're going to do for this one is we now need to turn, we're going to plug
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g of x into f of x.
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So I first need to turn, what is f of x?
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f of x is x squared.
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And now what I'm going to do is I'm going to plug g of x.
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So I do f of g of x into my other equation.
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So g of x is x plus 2.
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x plus 2 squared.
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Now for this problem, I can simplify this a little bit more.
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I can actually expand this out.
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And so I'm going to simplify, but just expand it out.
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So practice.
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Remember x plus 2 squared is x plus 2 times x plus 2.
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x times x is x squared.
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x times 2 is 2x.
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x times 2 is 2x.
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Add those together.
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You get 4x.
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2 times 2 is 4.
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So that is your review or introduction
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to composition of functions.