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For f(x) = - 1-* and g(x) = 2x2 +x+6, find the following functions. (fo g)x); b (g o f)x); (fo 9)(3);d. (g 0 f(3)(fo g)x) = (Simplify your answer)(g o f)x) = (Simpl...

Question

For f(x) = - 1-* and g(x) = 2x2 +x+6, find the following functions. (fo g)x); b (g o f)x); (fo 9)(3);d. (g 0 f(3)(fo g)x) = (Simplify your answer)(g o f)x) = (Simplify your answer:)(0 9)(3) =4. (g 0 f()=L

For f(x) = - 1-* and g(x) = 2x2 +x+6, find the following functions. (fo g)x); b (g o f)x); (fo 9)(3);d. (g 0 f(3) (fo g)x) = (Simplify your answer) (g o f)x) = (Simplify your answer:) (0 9)(3) = 4. (g 0 f()=L



Answers

For the given functions $f$and $\mathrm{g}$, find: (a) $(f \circ g)$ (4) (b) $(g \circ f)(2)$ (c) $(f \circ f)(1)$ (d) $(g \circ g)$ (0) $f(x)=3 x+2 ; \quad g(x)=2 x^{2}-1$

Hello. So here we have these two functions F of X is equal to X. G of X. Is equal to three X squared plus one. And report a we're looking to find F composed G of four. So we call that F composed G means F of G of X. Not writing it like this. We actually see what is our outside function and what is our inside function? So F composed G. It's just F of G of X. We take our outside function two X. And now we are in putting G fx in two. F. So therefore F of G of X. Well that's going to be equal to two but then times while we see it X we now input G. So that's two times three, X squared plus one. Okay, this becomes well two times three. We just distribute here. This is equal to six X squared plus um plus two. Right? And then we're now finding F this here is composed G. We're finding F composed G of four. That just means inputting now four into F. Composed G. That's just six times four squared plus two. So what's that? Four squared is uh 16 and then 16 times four is uh 64. This is a 64 plus two, which is uh 66. So there is F composed G of four. And then we're trying to find um G composed F of two. Okay, so now part B we're looking to find G composed F of to what is G composed F. That is G of F of X. Now our outside function is G. And now we're inputting in F. So we start with three X squared plus one and we see annex input F. So we have three times uh two X squared. So this is going to be equal to three times two x. There is our input for X. But then that squared, that's three X squared. Then we have plus one. So this is equal to while two X squared is four X squared. This is three times four X squared um Plus one. This is equal to one of the three distributes. And we see this equal to 12 X squared plus one. Okay. And now we're finding this. So here is G composed F. Finding G composed of two. Just need to put in now input to infer X. And we see this is equal to 12 times two squared, which is four plus one. So 12 times four is equal to uh 48 and then we have plus one which is equal to 49. So therefore we have that G composed F of two is equal to 49. All right. And then for part to see what we have, we have F composed F of one. Okay, so F composed F into inputting F into itself. So F part C. We have F composed F of one. F composed F that F of F of X, which is equal to two. Okay, But then we have what we input fx into itself. So we have to X. We have two times two X, which is equal to four X. And then we just put one in four X. So we have just four times one which is equal to four. So F composed F of one is equal to four. And um then we have party we have G composed G of zero. Okay, so G composed Gm zero. So G composed G is we put G into itself. So G we're going to find G composed G of zero. So G composed G is G of G of X. That's going to be equal to Well, we have three X squared, so we have three but then we have an X. We input in G which is three X squared plus one. We have three times three X squared plus one. Okay, But then that is squared because we input it into itself to have three X squared plus one. Now we have a by normal here we have to distribute out. So we have this G composed G is going to be equal to three times three X squared plus one times three X squared plus one. Let me have a plus one. Okay, so this is going to be equal to three times what we have a nine X squared. Right? The export empty squared, let me have a three X squared plus one plus a three X squared times one. Just three X squared plus three X squared, which is a six X squared. And then we have a plus one times one plus one and then another plus one out here. So now there's three is gonna distribute it's going to be equal to three times nine X squared. Which is going to be 27 X squared. And then a plus three times six X squared. That's going to give us an 18 X squared. And then a plus three times one plus three. And then a plus one. Which ends up being well we have plus one. Right? Which this just becomes then A plus four. Okay. Eight plus four. Okay. But now we're trying to find while G composed G of zero. So here's G composed G. But then in putting in zero would just give us while 27 times zero squared plus 18 times zero squared plus four, which is just equal to well zero plus zero plus four which is four. So therefore G composed G of zero is equal to four. All right, take care

He were composing some more functions and evaluating them. That given values another problem in four parts here. So part A is F of G four. So we're going to write G of four, and that's equal to three minus 1/2 times four squared. That's equal to three minus 1/2 of 16 which will be three minus eight, which is a negative five. That output of negative five then becomes the input for F in F of negative five is going to be four times negative. Five squared minus three or four times 25 minus three, which is going to be 97. Moving on to Part B. We do the same thing, but in reverse. This time we start with F, and we do f of to to get four times two squared minus three, which is four times four minus three, which is going to be 16 minus three or 13. We're then gonna plug 13 into G. I think it three minus 1/2 of 13 squared. That's going to be three minus 1/2 of 1 69 So using ah calculator, we can see 1/2 minus 1 69/2 is negative. 81 a half or negative 1 63 halves. Part C Moving on. We have FF so f of one is going to be four times one squared, minus three or four minus three, which is one We then plug in one again and we see that we get one out as well. Our last one is G of G of zero. So we'll plug in zero into G to get three minus 1/2 of zero squared. That'll be just three. That last term cancels and then g of three is going to be three minus 1/2 of three squared, which is three minus 1/2 times nine. So three minus nine halves is going to be a fraction that will be negative three.

Okay, so are two functions here are ffx equal to four X squared minus three, and G f x equal to three minus one half X squared. Now, in part A we would like to calculate F composed G of four, so that's going to be F of G of four. Um So that's F of while three minus one half times four squared. That's just F of three um -8, which is f of -5 which becomes Um so after 85, which becomes what becomes four uh times five squared minus three, which is equal to 100 minus three, which is equal to 97. Alright, so I've composed G of four for part B, we'll have to compute G composed F of two. So that's G of F of two. Now our outside function is G R. Inside functions we find ff two and then input that into G. So um F of two. Like the F that's four times 4 to 16 minus three, which is 13, therefore we're finding G up 13. So what is G of 13? Let's go ahead and input 3 13 into Gs we have three minus one half times 13 Squared which gives us um negative 163 over two. Mhm. All right. And then for see, we want to compute F composed F of one, that's F of F of one, defined F of one and input that into F. So F of one is four times one squared minus three. That's just four minus two, which is one. Let me take one and then we put it into F. But input into F just gives us again for minus three. So therefore F composed F of one is F F f of one, which is just equal to in this case, one. Um and indeed want to compute G composed G of zero. So G composed G of zero is G of G of zero. So you find G of zero and then put that into G G of zero is going to be just three, right? Because putting zero, this term goes to zero is three minus zero, is three. So G composed G of zero is just finding what is G of three, because Gm zero is 3 20 G of three. Well, that's three minus one half times three squared, that's three minus um one half times nine, which gives us uh negative three halves. So G of three is going to be equal to negative three halves. Okay.

So F. Of X equals X. The three halves G of X equal to two divided by X plus one. What we're gonna do is we're gonna find the composite function F. Of G. Of X when X is equal to four. So this is equal to F. Of G. of four. And so if we plug in four into our G. Of X equation we get two divided by four plus one or 2/5. So this is equal to F. Of 2/5. And so now if we plug into fifths into our ffx equation, we get 2/5 to the three have's power. And so if we plug this into a calculator to divided by five to the three divided by To the 3/2 power Is equal 2.25 2 9. So this would be equal to about .253 for part B. What we're gonna do is we're gonna find the composite function G of F. Of X. When X is equal to two. So this is equal to G. Of F. F. Two. And so if we go ahead and look at our fx equation, we plug into, we get two to the three halves. So we're going to have G. Of X. Where X. Is equal to two to the three halves. So we're gonna have G. Of 2 to 3 halves which is then equal to two divided by two to the three halves plus is a plus one. Yes plus one. And so now if you plug this into a calculator to to the 3/2 power And we add one And then we take two and divide that by the some we get about .52 two for part C. What we're gonna do is we're gonna find the composite function F. Of F. Of X when X is equal to one. So this is f. of f. of one and F of X was equal to X. The three halves and one of the three halves is just one. So this is equal to F of one, Just one to the 3/2 again, which is equal to one. Last thing we're gonna do is we're going to find G of G of X when X is equal to zero. So this is equal to G G F zero and G of X is equal to two divided by X plus one. So this is G of two divided by zero plus one. Just got a G. of two. And so now if we plug into we get two divided by 2-plus 1 which is equal to sorry not 32 3rd.


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