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Please g evaluate fhe iotegcal fx -xa A USiN6 Trlg Substtuion And (-Sobst #ution another Methodtkak Joj...

Question

Please g evaluate fhe iotegcal fx -xa A USiN6 Trlg Substtuion And (-Sobst #ution another Methodtkak Joj

Please g evaluate fhe iotegcal fx -xa A USiN6 Trlg Substtuion And (-Sobst #ution another Method tkak Joj



Answers

Compute $\frac{d}{d x} f(g(x)),$ where $f(x)$ and $g(x)$ are the following: $$f(x)=x(x-2)^{4}, g(x)=x^{3}$$

This problem were given to functions F of X equals two X gram plus one in G of X equals three access lanes. Five These multiple parts is problem in the first part part A. We need to find the value f composed of G of to all right G of two and red to be consistent with G of axe and red F in black so f of g of to this will equal the outside function of f of do you have to, which is simply g of access to as the input that becomes in red three times to minus five. This equals af, uh six miles, five on the inside or simply af of one as GF two equals one. Now we can plug in one for acts into f of X. So if that will get to times one squared plus one which is equal the two times one or two plus one or simply equal 23 f of G F two equals three in part B is notifying expression for f of g of axe. So let me scroll down part B. We need to find f of g of X as the input into effort backs this for eagle F of just that expression for G of X, which is three acts lines five you need plugging the hole three X minus five in tow Is this becomes to x squared? Or, in this case, that would be too. Times three X minus five Quanyou Squared plus one. We can simplify this out a little bit to get two times three X minus five times three x minus five plus one We can use foil on the to polynomial terms do you? Two times three extends three Access nine x Squared three X Times Mines Fire X minus 15 X on the outside, Then inside, we have mines five times three x Another negative 15 x It's a negative. 15 x minus 15 Accident mines 30 acts and then mines fired. Has minus five. Is positive 25 plus one Where this becomes too. I was nine. Sorry. No, I need to distribute the to teach part in the parentheses that becomes 18 x squared minus 60 acts plus 50 plus one. Finally combine like terms that becomes 18 x squared minus 60 acts plus 51. This what f of G of X looks like for part B Percy you to find expression. Fergie of epic fax G you right then red again G of F of X Being input into G looks like G of f of X f of X being two X squared, plus one expression with input into G. This becomes as G is three X minus five. Originally that becomes three times x or two X squared, plus one minus five, or for distributed three becomes six X square plus three minus five Really. Six X squared minus two is G ffx. It's going down. That's fine. Part D part do you need to find G composed of G of axe? This will be G off g of X on the inside, which is equal the G of G of X being three x minus five. The three x minus five is now input into G says really three times input of three X minus five minus five or for distribute will get nine x minus 15 minus five, which becomes nine acts minus 20. Fergie, composed of G of X and finally, her e. We need to find F composed of at of negative too that looks like F composed of half of negative, too, which looks like half of F of X is two x squared plus one bow plugging negative to forex. To get two times negative to square plus one is important. F This becomes half of think of two Squared is four in two times. Four is eight a plus. One brings us to nine Stanic unplugging nine into half get two times nine squared plus one, which is two times 81 plus one, which is 162 plus one are simply 163 is f composed of f of negative, too. So in summary, we went through five parts of the problem, find different compositions and found their values.

I'm not entirely sure how they want you to solve this problem. Um, if you want my opinion, I would actually figure out what f of g of X is just because I think this is better to understand this than to find some shortcut. So you take the G of X function, which is that six X minus one and you put it into the F function which is taken to the fifth power Eso Why is this a change? Rule is you have an inter function, the six x minus one and then the outer function is to the fifth power Because more problems were written in this than anything else. Eso then the derivative of that would be a prime. You leave gox alone times The directive of glx eso the outer function is is the what I'm talking about where it's five now to the fourth power. What you do is you leave that inter function alone and then multiplied by the directive of the inter function, which is six now. If you wanted to, you could simplify this. Some people wouldn't, but I would just go ahead because you can multiply that six in front to get 30 six x minus one to the fourth power

In this question, we have given you a function key. Exit of it is equal to minus X Esquire minus two eggs and bless. Okay. And let's say this is a question. First you have to evaluate the function at the are in the indicated values. OK, that means we have provided were dysfunction for the different avenues off access. And we have seen hell or what is actually given in this portion. We have given you a number. Excellent. And we have applied some Russo on this number Excell to find out the value of functions such as cakes. Okay, so we have to squired the number, the negative sign than my dependent to our and subtracting a two, then heading. But so we will find out here a venue or function K exit. So Ah, first of all, we have to find out the value of this function. R four x is equal to zero, so we have to put X is equal to zero in this equation first, so it will be equal toe minus zero square minus two in tow, zero plus tree. So after simplifications Avenue off K zero is equal to treat. Okay. Now, coming to the next. We have to run with the function X is equal Duke. So they will put extra support to do in this equation first, and after putting excessive called toe, it will be equal to minus four, minus school plus three. So after simplification, he can say they're double your conscience. Texas equipment to be equal to minus five. Okay, now coming to the next one here, we have to evaluate the function it x is equal to minus two. So when we put here accidents going to minus two, it will give The desert is minus four and plus four plus So passport and minus four begins allowed and the value of function K minus true will be equal to three. OK, now coming to the next one. Also, we have toe evil way to here. OK, let's see excess tickle toe. So when we put X is equal to go through in this function and so according to the question first, it will be called to after putting excess tickle toe to minus two minus toe plus three. Okay, so from here because say that the real the function at X is equal to two will be equal one minus will. Okay, now, coming to the next one. So in this valley, we have toe even week. The function act A plus two. That means we have to find her the value off K A. Plus two. So according to the question, first, it will be like this minus a plus two. Holy square, then minus two A plus two and plus. Okay. Now we have to simplify eight. And after simplification that miss after. Oh, applying here. Identity is a spur place. Beast. So the A plus B equals square and with negative size, or you'll be minus X squared minus sport minus four E and minus for a minus four. Blessed. Okay, now we have to simplify eight. So, after simplification here, with the value of the function at K personally I takes is equal to a plus two will be equal minus ace quiet and minus eight e minus free. Okay, Now, the next value, which is giving here in this portion O is minus x it. Okay, so we have to evaluate off. Function at X is equal to minus six. So we have to replace X by minus excellent in this equation first. Then it will be equal toe minus x Esquire and bless two eggs. Blustery. Okay, so you can see that of envy. Replace expert minus X. So this is excess choir. So this will not affect too. But this will be in positive sign here. Okay. Coming to the next year. So in this puncture, we have to evaluate the value off function at excess square. So we have to kowtow. Okay? I'm just replacing it. So we have to put X is equal to X Esquire in this occasions cost. So you can understand when I put excessive all day to square, even be equal to minus six to power food minus do extra square and plus two. Okay, so this is the value of a function at X squared. So you from here we can say that we have find out all the values for the different values off, except for dysfunction. The 1st 1 waas k zero, then Kate too. Then K minus two. Then kill you too, and then gave a plus two. Okay. Also, I'm just replacing it. K minus k. Please do. Then Q minus X and then k X squared. So these are all the answers for this question.

Okay. So here we have that are functions F and G are defined by older pairs. And we're gonna find here G composed G of six. We're G composed G is equal to this is G of G of six to find G. 06 and then input that Mtg. So what is G six or G of sex is equal to zero? Right? Because um G of sex when the input is six and G the output is zero. So G F six is equal to zero. So we're finding then is well G of zero. Yeah. And what is G of zero? G of zero is six. Because um in front again in G when the input is zero, the output is six, so G of zero is equal to six. Therefore G composed G of six is equal to six.


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