In Exercises 51–66, find
a. (fog) (x)
b. (go f) (x)
f(x) = 4-x,...

Question

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x) 
f(x) = 4-x, g(x) = 2x² +x+5

Accepted Answer

So this problem says to solve F of G of X and G F F of X where f of X equal to four plus squared of X and G of X is equal to X squared plus two, X plus two. Let's first work on F of G of X. Right down here, F f G X. That is the same thing as saying F on the outside and two on the inside. So we're gonna take what we have for T F X and substitute in for the X and r f F X equation. So feel like here F X is going to be F of G of X X squared plus two X plus two. No, we went to rewrite party question. So the question is four plus squared of X. So we have four plus the square root of our G X squared plus two, X plus two. Now we can't really simplify that polynomial any further. So we're going to leave it as it is. Think that that equation it's four plus squared of X squared plus two, X plus two. Now we'll do job X over here, G of F of X. Now our G is X squared plus two, X plus two. We'll say G four plus one of X. That's going to be equal to four plus squared of X squared plus two times four plus squared of X plus two. Now we want to simplify that simplify it. We have four plus squared of X squared. So four plus squared of X times four plus squared of X. Now we have two tons. Four plus square root of X distribute are to be able to get eight plus two squared of X And plus two just comes down. Now I'm gonna multiply these two polynomial here, which we can do with foil. So multiply. Our first terms are outer terms, our inner and our last add them together. We will get four times four which is 16, 4 times a squared of X is four squared of X plus another four sets of X. Finally squared of X. Some squared of X was just X. Then we add eight's in our two squared and X and R. Two. Now I can send the fire even further by combining like terms Look here we have a 16 and eight into two, combining all of those gets us 26. We have a four squared of X plus a four squared of X and a two squared of X. Add all those together And we get 10 squares of X. Finally you know it's here, we just have a single X. Here. So we just bring that X down. So our G. Of F. Of X equal to 26 plus 10 squares of X plus X. Combining that with our answer for F. Which is four plus squared of X squared plus two, X plus two. We get answer. A Okay, I hope to help you solve the problem. Thank you for watching. Goodbye

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