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For the functions f &d given bclo find the composite functions f and & #f . and the domain each:fW) =k-61 g(x)Enter - vaur anseensfully simplified and expan...

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For the functions f &d given bclo find the composite functions f and & #f . and the domain each:fW) =k-61 g(x)Enter - vaur anseensfully simplified and expanded formn Enicr the domains intetval notation: Enclose numerators and denominators Pireninenes #umnple (a - by (+n) or (a ~ bV (a *x)To enter |x/: type abs(*)To enter co , type infinityTo enter Ihe union symbol, type UInclude multiplication sign betwcen symbols, For cxamplo: (a 1b *yU( %d*2DomainDomain -elkhnculj Ulvg = Shoxs Wat dar

For the functions f &d given bclo find the composite functions f and & #f . and the domain each: fW) =k-61 g(x) Enter - vaur anseens fully simplified and expanded formn Enicr the domains intetval notation: Enclose numerators and denominators Pireninenes #umnple (a - by (+n) or (a ~ bV (a *x) To enter |x/: type abs(*) To enter co , type infinity To enter Ihe union symbol, type U Include multiplication sign betwcen symbols, For cxamplo: (a 1b *yU( %d*2 Domain Domain - elkh nculj Ulvg = Shoxs Wat dar Ulo guedon Qeen_ShoiWod J*8 8'f



Answers

For the given functions fand $g$, find: (a) $f^{\circ} g$ (b) $g^{\circ} f$ (c) $f \circ f$ (d) $g^{\circ}$ g State the domain of each composite function. $f(x)=x^{2}+1 ; \quad g(x)=2 x^{2}+3$

So here we have these two functions F of X equal to X plus one, and G F X equal to X squared plus four. So for part A here, I want to find F composed G So F composed G. That means F of G of X. Where our outside function is F. And we are in putting G. So we go to every backs now, wherever we see annex, we input Gm X for our input anthrax. So therefore um F composed G wow, that is going to be equal to we have an X squared plus four that were in putting in for X. And then we have a plus one. Well X squared plus four plus one. That's just equal to X squared plus five. Right? This is just a parabola, X squared up by five units. So our domain, right, is all real numbers right? Put in any value for X. You get an output so that the domain again is all real numbers. Okay? Um And then part B We have G composed F. So G composed F. That means G of F of X. Now our outside function is now G now we're inputting N. F. So F composed G. Well we have an X. We input F. So we have X plus one. That's our input. We have it squared, right? This X squared plus four. So um again we could I guess I mean really you want to leave it like this? You could or you can clean it up and say okay X plus one squared. That means X plus one times X plus one. Then we have a plus four. So X plus one times X plus one gives us an X times X which is an X squared. And then X times one is X plus one times X. That X plus X plus two X. And then plus one times one plus one plus four. Give us a plus five. So either X plus one squared plus four, which is equivalent to saying X squared plus two. X plus five. The domain again? All real numbers. All right. So there's uh F composed G and G compose F. Then for part C. We're looking at F composed F. So that means if of F of X. So now we're inputting F into itself. So our function F is X plus one. So therefore we have X plus one is our input. And for X and then we have a plus one. So X plus one plus one, that's just equal to X plus two. And domain Argo numbers. All right. There's a C. And then D we're looking at G G composed G. So that means G of G of X. So our function G is X squared plus four. So we're inputting G into itself, so therefore we have X squared plus four is our input for X. We have that squared and then we have a plus four. So there is um There is G composed G if you want to go ahead and again square this out and then combine in terms because if you want to, but if we want to just like this, that's fine too. Um And again the domain, all your numbers.

Okay, so I function here are fx equal to three X minus one and G f X equal to X squared. So let's first part A. So I guess part A here is going to be F composed F. Remember that F composed F? That means F of Oh, that F composed of. Well actually, hey A is actually composed G right composed G First F composed G F composed G means F of G of X. So we have F of G of X. Okay, so our our outside function is F and we're inputting G wherever we see annex in our function. So we have three. Well, x minus one. So we have three X. So therefore our input is G which is X squared and then we have minus one. So therefore it's gonna be equal to just three times X squared history X squared and then in minus one. Um Again the domain here. All real numbers right. You can put in any value for X and get an output. So all your numbers negative, negative infinity. To infinity would be our domain. Okay. And then the part B let's do now we did have composed G. Now let's do G composed F G composed F. That's that means G of F of X. Okay, so G of F of X. Our outside function is now G which is X squared? So whenever we see an X, we input now F. So we have three x minus one squared. So we have three x minus one. Is our input for X squared? Right? X squared. Our input is um F Which is the x minus one. So what does this mean? Three x minus one squared? Or that means three x minus one times itself. So times three x minus one. Okay, now we distribute. So you have three x times three x. That gives us a nine x squared. They have three x times negative one minus three X. And another minus three X. That's a minus six X. And then uh minus one times minus one plus one plus one. And so here we have it. So um so G composed F is equal to nine X squared minus six X plus one. Uh domain. Again, this is just a parabola, right? The domain is again all real numbers. Okay? Um And then for see we are now doing F composed F. Now we're inputting F into itself. So F composed F means F of F of X. So we look at our function f which is three X -1. So we have three but then X. R input is F. So it's three x minus one, so it's three X. And then minus one. So this becomes three times three X. Is a nine X. And then three times minus one is a well minus three and then we have the minus one. So that's nine x minus four. Okay. Again, the the domain all real numbers Okay, There is composed of And then for D we have G composed G. So now we're inputting G into itself. So G composed G. That means G of G of X. So we look at our function G which is X squared. So we input X squared into itself. So we have X squared in for X. But then squared while a powerful power is we multiply our powers. So X squared squared is just X to the fourth. And the well the range right? It's only positive here. The domain is all real numbers. All right. Take care.

Okay, so we're going to find a couple of composite functions involving these two um functions F of X and G of X. First one is F of G of G of X. Which is just we plug in G fx in for X. So this will be equal to one divided by negative two divided by X plus three. And so the domain of this function is going to be the range of our G fx function. And the only thing is we can't have this negative two divided by X equal to negative three because then we'd have a zero in the denominator. So the two places that we can't have are the two things that we can't have is we can't have a zero and this denominator and then we can't have a zero in this um whole denominator down here. So we can't have X equaling zero. So X cannot equal zero and we can't have negative two divided by X plus three equaling zero. So we can solve for X here u minus three over negative two divided by X is equal to negative three. So negative two is equal to negative three X. And so X cannot equal um negative two divided by negative three is positive two thirds. So we can't have X equaling two thirds and we can't have X equaling zero. So our domain is going to be from negative infinity 20 and then From 0 to 2/3 and then from two thirds to infinity. And all of these brackets on the zero and the two thirds should be parentheses and brackets. Since we're not including those numbers for part B, we're gonna figure out the composite function G of F. Of X. So G. Of X was negative two divided by X. So we have negative two divided by we plug in F of X for X. So we have divided by one divided by X plus three. And so here we can see that we can't have zero in this denominator but we also can't have zero in this denominator. And so we're actually never going to be able to get um zero in the denominator. Everyone talked about negative two divided by one. Um divided by X plus three. We're never gonna have zero and that's nominator since one divided by X plus three can't equal zero but we will have zero in this denominator so we can't let that happen. So X Is not equal to -3. Um is the domain, it's all real other than -3. So you could say from negative infinity two negative three. You from negative three to infinity is our domain part C. We're gonna find F Fx And so F of X was equal to one divided by X plus three. So this is equal to one divided by one. My bags plus 3-plus 3. And so here we can't have again X equals negative three. So we could have a zero and this denominator and we also can't have one divided by X plus three plus three. We can't have this equaling zero either. So we can minus three over. We get one divided by X plus three equals negative three. So then we get one is equal to negative three X minus nine. And I just distributed this negative three through this X plus three when I multiplied it on both sides. So now we can plus nine over 10 is equal to -3. X. So X would be equal to negative 10 divided by three. So we can't have X equals negative 10 divided by three because then we would have a zero in this denominator. So our domain is going to be from negative infinity. It's a -10 divided by three. And then from negative 10 divided by three, It's a -3. And then from negative three to infinity since we can't have negative 10 divided by three. And because we'll get a zero and this in this denominator here and we can't have negative three because I'm going to have a zero in this denominator here and this was part C surround part. Uh The last thing we're gonna do is we're gonna find G of G of X when X. Or just yet. We're going to find G of G of X. So G of X is equal to negative two divided by X. So this is equal to negative two divided by negative two divided by X. And so here again we can't have X equaling zero since we'll have a zero in this denominator here. And other than that, we can have X equaling any other real numbers. Since this negative two divided by X can't be equal to zero. So we will never have a zero in this whole denominator here, but we will have a zero in this denominator if we let X equaling zero, so our domain is going to be from negative infinity 20 And then from 0 to Infinity.

Okay, so our functions here are F of X equals two X squared, and G f x equal to X squared plus four. So for part a looking to find F composed G recall, that means F of G of X are outside function is F and we're inputting in G and F. So therefore this is just going to be equal to well we have X squared. We whenever we see an X going to input in G, so we have X squared plus four, there's an input and that is just squared. So there is um F composed G if we wanted to we could go ahead and square this out. We have X squared plus four times X squared plus four and distribute. Um and we would see that the domain here would be well, all real numbers. Okay, so there is F composed G. And then for be looking to find G composed F recall, that means G of F of X. Where now our outside function is now cheap. Okay, So we go to G which is X plus four and we input in F which is X squared? So therefore we have X squared squared. Right? There is X squared R input for X is if X then we have a plus four. So there is X squared um plus four. Okay. And while X squared square, the power to a power is we want to play our powers, so therefore this is gonna be equal to X to the fourth plus four. Okay, So here is um G composed F And we see that you put put in any value for X here um for an input and get an output. Therefore the domain all the possible inputs would be all real numbers. All right, okay. And then for C we're looking to find F composed F. Okay, well, that means we have um this means F of F of X where input F into itself. So I function F is just X squared. So therefore um F compose F is just going to be, Well X squared is our input in for X. So we have X squared or X squared squared is just equal to X to the fourth. So there is um F composed F and again the domain, there is all real numbers. And um for D we have G compose G, we are inputting now G into itself. So we have G of G of X. So that's going to be equal to well G is X squared plus four, so therefore we have X squared plus four as an input in four. Um and for X so we have X squared, let me have plus four. So there is um G composed G If you want we can, you know, if you want to uh clean this up, you could buy just um distributing this. So we would have X squared plus four times X squared plus four. Right? That's what X squared plus four squared means. And then we have a plus four. Well, it's going to be equal to X squared times X squared is equal to X to the fourth. And then we have a X squared times four is a four X squared plus another four X squared, gives us a plus eight X squared. And then we get a plus, well four times four is 16 and then we have two plus four. So therefore this is equal to X to the 4th plus eight X squared plus 16 plus four is 16 17 18 1920. So, um this right, if you wanted to leave it like this, you could or that becomes extra the 4th plus eight X squared plus 20. So that would be G composed G. And again, any value in the input here gives you an output. Saudi domain again, is all real numbers.


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