Let F(z) = f(z5) and G(x) = (f(z))5 You also know that a4 15, f(a) = 3, f' (a) = 8, f' (a') = 9Then F ' (a)and G' (a)...

Suppose $F(x)=g(f(x))$ and $f(2)=3, f^{\prime}(2)=-3, g(3)=5,$ and $g^{\prime}(3)=4 .$ Find $F^{\prime}(2)$

For this potion we have ffx it closed toe three X squared minus one and geo affects way have to express life. The first part of the solution. We have to find their fourth Z off minus four. So before that, let's say we're saying the air Force Z off excess work, so f d off Exeter's three times Z affects Holy square minus one. That is three times Dfx. We have to expose five all the square minus one. So f off G off minus four we have putting the value of X minus four That is three times two times minus off four plus five divided by holy square minus one, which is three times minus eight plus five minus off street holy square minus offline which is or three square, that is nine times three is 27 minus what is two interstates? So in this case, 26 were getting Which is the answer No part b of the solution we have f off that is zero for minus four. So before that, what we can say z off effects in sports that is Z off affections to times effects. Bless five that is two times if access three X squared minus one. Let's Fife, which is six x squared minus to plus five s three here Z off F off minus four. We should put the value of X minus for six times minus four. Holy square minus street that is close to fourty square. 16 times six is 96 minus three. So it is 96 minus 93. So 92 correct answers for this case.

Suppose you have F defined as the composition of F and G. And in here we want to find F of X and G of X when our F of X is equal to the cube of one plus X squared. Now F of X which is equal to one plus X squared grace through the third power. This tells us that one plus X squared is the inner function and the base function or the outer function of this would be X. Rays to the 3rd power. Now, if F is the composition of F and G, then F of X, which is F of G of X. This is the same as F of G of X, in which the inner function is G. So this must be G and F of X is the alter function, so a Fedex will be execute. And so we have these functions F of X equals X cube, and G F X must be one plus X squared.

Using the given functions, we need to find G of X. G of F. Of X and F of G of X. So for our first function we're going to replace N. G. We're gonna replace all the exes with what F. Of X is. So two times instead of X, we're going to replace it with three X plus five. Then we have -7. So we distribute until we get six X plus 10 minus seven. So this simplifies to be six X plus three. So now we're going to do the opposite. We're going to take the G. F. X and replace it into the F. Of X. So we have three times Times. So instead of X We're going to replace it with two X -7. What is the color 2? X -7. Now we have plus five. So we distribute we get six x minus 21 plus five and this becomes six X 21 negative 21 plus five becomes negative 16.

Giving the following functions. We need to find G of F of X and F of G of X. So our first one we're going to replace in G. Fx all the exes with F of X. So we have three Parentheses -5. So inside the parentheses we're going to replace with two X plus one mm. So this becomes six over X plus one minus five. So we're gonna multiply by the common denominator. So we get six minus five X minus five over X plus one. This simplifies to be 1 -5 x Over X-plus one. So now for the 2nd 1 we're gonna replace G fx for all the exes in ffx. So we have to over instead of X we're going to replace it with three x minus five. Then we have a plus one. So this becomes to over three X -4.