WEBVTT
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Let's start by finding f of G. So G
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is the inside function. So we're going to take
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one minus four x and put it inside the F
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function in place of X. So that's going to
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look like one minus four X quantity cubed minus two
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. So now if we want to simplify that,
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we're going to have to cube the one minus for
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X. So what will that look like? You
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might know a shortcut if you know the binomial theorem
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. If not, you're going to have to multiply
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it out the long way. One minus four X
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times one minus four X times one minus four x
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So if we do the 1st 2 using the foil
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method, we get one minus eight x plus 16
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x squared and then we're going to multiply that by
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one minus four x. So starting with the one
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and multiplying it by both terms, we have one
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minus four X and then moving to the negative eight
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x and multiplying it. By both terms, we
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have negative eight x plus 32 x squared and then
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moving to the 16 x squared and multiplying it.
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By both terms, we have 16 x squared minus
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64 x cubed. Now we can combine the light
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terms and we have one. We have a couple
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of like terms here These X terms so that would
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add up to minus 12 x have a couple of
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, like terms here the X squared terms and that
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would add up to 48 x squared and we have
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minus 64 x cubed. So let's go ahead and
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put that into the previous step. So we have
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that here and then we still have the minus two
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at the end. And so then we can combine
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the one in the minus two. So we end
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up with if we want to put the terms in
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descending powers of X, we end up with negative
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64 x cubed plus 48 x squared minus 12 X
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minus one. Okay, now let's find GF so
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f goes on the inside. So what we're going
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to do this time is we're going to take F
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, and we're going to substitute it in for X
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in the G function. So that's going to look
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like one minus four times the quantity X cubed minus
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two. That's a lot easier to simplify. We're
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going to distribute the negative four and we have one
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minus four x cubed plus eight. And then we
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can combine the one and the eight. So we
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get nine minus four x cubed. Now let's talk
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about domain. So looking at the F function,
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it's a polynomial. So it's domain is all real
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numbers like every polynomial is domain. Same with G
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. It's a polynomial. Its domain is all real
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numbers. F of G is a polynomial to it's
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domain is also all real numbers. And if you'd
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rather you can say that is negative Infinity to infinity
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Same with GFF. It's a polynomial. Its domain
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is all real numbers. Okay, For part C
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, we're finding FF, So we put the F
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function inside itself for X. So that's going to
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look like this. X cubed minus two cubed minus
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two. Okay, so here we go again.
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Cubing a binomial. We can work that out.
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X cubed minus two. Cubed is X cubed minus
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two times X cubed minus two times X cubed minus
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two. Same process we did before. Start by
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multiplying two of them using the foil method and we
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get X to the sixth power minus four. X
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quit cubed plus four. And then we're going to
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multiply that by the other binomial, starting with the
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1st 1 multiplying it by both. And we get
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X to the ninth power minus two x to the
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sixth Power and then moving on to the 2nd 1
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and multiplying it by both. And we get minus
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four x to the sixth power plus eight x cubed
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and then moving on to the 3rd 1 and multiplying
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it by both. And we get four x cubed
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minus eight, and then we'll combine like terms and
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we have extra the ninth minus six x to the
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sixth plus 12 extra, the third minus eight.
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So let's copy that back in the problem we were
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just doing and we still have minus two after it
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. Okay, so if we combine the like terms
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, we have X to the ninth Power minus six
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x to the sixth power plus 12 x cubed minus
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10. Now it's find GMG, so we're going
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to take the G function and substituted into itself.
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So four x minus one all goes in for X
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, so that's going to look like this. Four
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times a quantity for X minus one minus one.
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This one's a lot easier to simplify. Weaken,
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Distribute the four and we get 16 X minus four
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minus one. So that's 16 X minus five.
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And now let's finish up with the domains. So
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the domain of F of F remember the domain of
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F was, um, all real numbers, because
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F is just a polynomial, and f f is
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also just a polynomial. So it's domain is all
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real numbers, which we can write as negative infinity
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to infinity and same with G A. G.
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It's a polynomial, so it's domain is also all
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real numbers.