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Suppose H (x)=Sx 6. Find two functions f and g such that (f %) (x)=H (x) Neither function can be the identity function. (There may be more than one correct answer.)...

Question

Suppose H (x)=Sx 6. Find two functions f and g such that (f %) (x)=H (x) Neither function can be the identity function. (There may be more than one correct answer.)

Suppose H (x)=Sx 6. Find two functions f and g such that (f %) (x)=H (x) Neither function can be the identity function. (There may be more than one correct answer.)



Answers

Find two functions $f$ and $g$ such that $h(x)$ can be expressed as the function indicated. Several answers are possible. $$h(x)=x^{2}+7 x-18 ; f \cdot g$$

So here we had two functions that is affects Izabal toe Xs Square minus one, and GX is according X minus two, and we have to find the value function F by G at minus five. So, firstly, we'll find out the function of Baiji off X asked by G off eggs that will be quarto X squared minus one, divided by X minus two where X can or three required to do. Because this will make our function nor define. Now we will put the value off eggs. Is it going to minus fired and I function. It means minus five square minus one. Divide by minus five minus two. That is a quarto 25 minus one, divided by minus seven. It implies minus off 24 by seven. Which means F B I. G at minus fire is a quill Do minus 24 by seven that is equal to seven. Threes are minus three point and then fours. Ah, don't So a lot of value. Ah, function of my gear minus five is equal to minus 3.4. Do we can also verify our value by using profitability of a function Waas Xa square Our function. Waas X square minus one, divided by X minus two. So this is the graph or other function? F B i g. You know, we've been just verify our value at minus five and you can see that the value of Isaac will do minus 3.4. Do so. Our value is inquiry.

Okay, so we're given h of X, and we want to write each of X as a product of F. G. So that is why don't we let halfbacks be equal to three and then g of X be equal to X squared. So then we have a jukebox is equal to ever back times to your bucks and that you go to three times X quick.

So give it a function H. Of X. I want to find what F. And G could be so that when I plugged them in their composite functions. Okay so there's a couple ways to think about this. The way I think about this is I look at my innermost term, where is my innermost term? Some people say my innermost term is six X -2. Okay, so that's going to be my inside function. So I'm gonna say G is going to be six X -2. And then when I take away that inside function, what am I left with? What I'm just left with something squared Or in this case that something could just be X squared. Then when I plug this back in right, if I do F G fx then I'm replacing the entire G function six x minus two in for X. And I'll get sick the quantity six x minus two square. Another way to think about this. Some people would say that the innermost function is just going to be six X. So let's say I pulled out six X. What am I left with here? Something -2. All of that squared. Now, I'm going to replace the X value and F with G. So I'm just gonna put an X. Placeholder X here Fergie. This is another way to break this down and so on. There's a couple of different ways to solve this. But the important point is my G FX needs to have an X with it and I need to put a placeholder in for X where my G of X is going to be pumped in.

The function given to us in this problem is a job includes do it last one. Honestly. Now, to write it as a composition of two different functions, we can see there is a whole square operation done. And inside it we can see that the squared operation is done on two X plus one. So what we can right here this weaken right function equals two X plus one and then we can write effort X squared. So now if we, like, have composition D on a, this will be it. Well, you know, we can see geographic is licks this one and then if is a square function quit. So the valuable here will be two x plus one that will be to express one behold square. Therefore, we can write a topics as f bomb police nd on. Hey, that fo fix is explain. And Geo Texas Lex best one


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