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2 Find the function.1fkx) =x2 4x a. (g 0 f)(x)g(x) = x -5b. (f 0 g)(x)...

Question

2 Find the function.1fkx) =x2 4x a. (g 0 f)(x)g(x) = x -5b. (f 0 g)(x)

2 Find the function. 1 fkx) =x2 4x a. (g 0 f)(x) g(x) = x -5 b. (f 0 g)(x)



Answers

Given $g(x)=x^{4}-x^{2}-3,$ find the function values. a. $g(-1)$ b. $g(2)$ c. $g(0)$

So we're given The F of X is equal to three experts to and that G of X is equal to two X squared minus one, and we were asked to find various values of F G and G F perspectively the 1st 1 FG afford. We're going to plug foreign DeJean and then put the result in tow. So G of four is equal to four squared or 16 times two is 30 to minus. One is 31. Sergio four is 31. We're going to plug that into FF of 31 is three times 31 or 93 plus two, which is 2095 for the next one G of F of two were going toe to toe f. So half love too, which is equal to eight. And we're gonna plug eight into G so G of AIDS is equals. Eight squared or 64 times two is 1 28 minus one is 1 27 graph of f of one. We're going to pluck F into one and then put the result back into F F off. One is equal to five, so that F of five is equal to 17 the last one g of G of zero Well, G of zero is equal to negative one. So then G of negative one is equal to one.

So we're given that f of X is equal to two x clear and that g of X is equal to one minus three x squared harass, defying various values of f of G or G of F. So for the 1st 1 we given effigy of force or going to plug for into G and then pluck the result in tow. So g of four is he got a one minus three times 16 which is one minus, um, 40 eight's that is going to be negative. 47 and we blood negative 47 in for F and that is going to be equal. Teoh 47 squared, which is 2209 times two is going to be 4418 for the next one. We're going to be doing gff of twos we're gonna take after to unplug, didn't itchy well f of two is equal to eight and then g of eight is equal to one minus three times 64 which is going to be equal. Teoh Negative. 191. The next one f of f of one. We're going to plug one into left and then let the result back in tow. So one into F is equal to two, so f of two is equal to eight. Then finally, for G of G of zero, we're going to plug zero into G and then plug that back into G zero. Fergie is one minus zero or one g of one is one minus three or negative, too.

So we're given that f of X is equal to four X squared minus three, and that G of X is equal to three minus 1/2 x squared. They were finding find very spires of F of G or G of F. So the 1st 1 f of G four, we're going to plug four into G, which in this case is negative five and then plus negative five into F left of negative five is going to be equal to five square or 25 times four is 100 minus three. That tells us that F of G of four is 97 for the next one G of F of to well F of two is equal to four times for just 16 minus three, which is 13 and then G of 13 is equal to 13. Squared is 1 69 Then it's three minus half of that which is going to be 84.53 minus 84.5. This negative 81.5 for the next one f of f of one well F one is equal to four minus three, which is one, and we just calculated off of one. It is one. It's 1/2 of Africa born is one. Finally, drastically GMG of zero g of zero is equal to three, and then G of three is equal to negative 1.5.

Hello. So here we have our functions F of X equal to three X plus two and G of X equal to two, X squared minus 1.5. To find what is F composed G of four. Well, first let's find let's find what is F composed G. And then input four into F composed G. So what is F composed G? That just means F of G of X. So F composed G would be F of G of X. Okay, so that's going to be equal to we start with F of X as our outside function. So we have three times X. But X is our input which is now G of X. So we have three times two X squared minus one and then we have a plus eight plus two. Okay, that's going to be equal to three times two. X squared is a six X squared and two times minus one is a minus three. Then we have a plus two. Right, so minus three plus two is a minus one, so therefore we have six X squared minus one. Is F composed G. Um Now if I have composed of four, just go ahead and put in put four into F. Composed G so F composed of four is just six times four squared minus one. That's equal to six times or fourth quarter 16? That's six um times six times 16 um minus one. Uh So six times 16 is ah what is that? Six times 16 is 96 So um 96 minus one, which is equal to 95. So therefore F composed G of four is just equal to 95. Okay. Um And then for part B we are asked to find um G composed F of two, So G composed F of two. So first let's find what is G composed F. That's G of F of X. Okay, so that's going to be equal to our outside function is now G. We have two X squared. So we have two times X. What is X? X. Is now F. Which is three X plus two. So we have two times three X plus two squared, right? There's two X squared. And then we have a minus one. Okay, well we have to do some distributing here. Three X plus two squared means three X plus two times X plus two. So therefore this is equal to two times three X plus two times three X plus two. And then we have minus one. Okay, let's go ahead and distribute or foil out these two binomial. And then everything is gonna be times too. So we have two times while we have a three x times three X. That gives us a nine X. Squared. And then a three X times two X. Gives us a six X. Plus another six X plus a 12 X. And then it plus eight times two plus four. And then we have the minus one. So this is going to be equal to uh we have a 2009 X. Squared, which is an 18 X squared. And then we get it two times 12 X plus 24 X. And plus uh damage for which is a plus eight. Then we have a minus one. Right, so eight minus one is going to be seven plus seven. All right. So there is um what is that? That would be G of a fax. We're gonna find G. Of F G composed of two. That means we just put now input too. So we have 18 times two squared 18 times four plus 24 times two plus seven. Okay, so whatever this works out to, that's going to be 18 times four, which is 72 then plus 24 times two sets plus a 2030 40 48 so plus 48 we have 72 plus 48 which is 120 then plus seven which is 127. So um we have 127. So that's going to be G composed F of two. G. Composed ff two is 127. Okay, um Next what we have, we have G F. Composed f of one. So F composed F of one. That's gonna be F composed F of one. That's F of um F of F of X. Is F composed of and then we will be input in one. So what is F composed F? Well, F composer F F of X is three X plus two. So we have three X plus two is F right? Three. But then X we put me input well F into itself, so three times three X plus two. And then A plus two. Right, that's three X plus two. What's this going to be equal to that? Three times three, which is nine X. And then three times two is a six, so plus 12 plus six plus two, gives us a plus eight. So we have nine X plus eight. Now we're inputting one. So we just have um nine well, times one plus eight, which is just nine um plus eight, which is 17. So therefore we have F composed of one is equal to 17. And lastly for D we have G composed G right of zero. So what is G composed G of zero? G. Composed G means G of G of X. We have two X squared plus one. Right? But we have one of mine G of G of X. So G is to exclude plus one. We have two times while two X squared plus one uh squared right? That's two X squared um minus one minus one minus one. Right? That's two X squared minus one. Right? Two X squared and then minus one. Okay, so what is this equal to? Well this is equal to two times 12 squared minus one. Get squared. So we have two X squared minus one times two X squared minus one and minus one. Okay, so what we have here we have this equal to two times two X squared times two X squared is a four um acts to the fourth and then two X squared times of minus one. The minus two X squared and then another minus two X squared to minus four X squared minus one times minus one is a plus one and then we have minus one. So two distributes this is going to be equal to um eight X to the fourth and then in minus eight X squared. Um And then a plus two minus one becomes just a plus one. Okay then inputting zero right, this term goes to zero, this term goes to zero and therefore we get that G composed G of zero is just equal to well one. Take care.


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