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Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = x^3 - 2 $ , $ g(x) = 1 - 4x $

a) $=-1-12 x+48 x^{2}-64 x^{3}, x \in(-\infty, \infty)$

b) $=9-4 x^{3}, x \in(-\infty, \infty)$

c) $=x^{9}-6 x^{6}+12 x^{3}-10, x \in(-\infty, \infty)$

d) $=-3+16 x, x \in(-\infty, \infty)$

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