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Suppose that the functions h &nd f are defined as follows #G)er f6)=Fnd tht compositions h %h and f % simolty your ansers as much &5 possible: (Assume that ...

Question

Suppose that the functions h &nd f are defined as follows #G)er f6)=Fnd tht compositions h %h and f % simolty your ansers as much &5 possible: (Assume that your expressions are defined for al x In the domald of the compositlon; You do not hove Indicate the domal )"96) = @6*6)a0

Suppose that the functions h &nd f are defined as follows #G)er f6)= Fnd tht compositions h %h and f % simolty your ansers as much &5 possible: (Assume that your expressions are defined for al x In the domald of the compositlon; You do not hove Indicate the domal ) "96) = @ 6*6)a0



Answers

In Exercises 75–82, express the given function h as a composition of two functions f and g so that $h(x)=(f \circ g)(x)$.
$$h(x)=\frac{1}{4 x+5}$$

So we're given this function over here more asked to, um express the given function age of the composition of two functions F N g. So that each of X is equal effort. You vics. So we want h of x Toby expressed of the composition of two functions such that h of x is equal f of X half of geophysics. Basically, we want to find ah, function. That's gonna be the what? I did it again. Sorry, it's gonna we're looking for and f of gox and so basically won't want a function that's considered the outdoor function, after which we can put this inter function. And we want this over here, this whole h of X to be equal to this thing over here. And so let's let's look at this function What can be the input to the function? If you think about it, it makes sense that two X minus five is the input to this function. Or actually scratch that. Let's talk about what the outer function will be. The outer function is gonna be your f It's gonna be what your input goes into. And just by looking at this, we can see that our outer function f of X is going to be eggs is going to be X cubed. Any makes sense because this X over year is gonna be where our import get substituted, which is the case here. So if this is our outer function, our inner function, it's only logical if our inter function is this thing over here. Two x minus five. So now if we do f of g of X, basically, wherever I see an X in my half of its function, I'm replacing it with the G of X, so it's gonna be equal the two x minus five cube. Um, it's it's kind of tricky efforts, but you just have to recognize that whenever you have these kind of functions, like, say, you have some expression year to the end power, always when and if you want to express as a composition, always treat this outer function like X to the end as your um as your f of X or as your outer functions heart and the expression is always going to be your inner function or G of X. So it's kind of tricky at first, but you should have to learn to recognize what the outer function will be and what the inter function will be. And so I hope that stuff.

So we had a chest X is equal to 1/4 X plus five, and we want that function h of X to be the composition of two functions, which I like to write as f l g of X. And so we want to know what does Alpha Vax have to equal and what does g of X have to equal? And when I write it like so I can see that this is acting as my inner function and I see that four X plus five is acting as an inter function now, I think. OK, well, then what is the EP function, which is the outer function? What is that doing? And that outer function is taking one over the input and the way I have written X is my input. So there's my app and there's my G. And if I want to check, I could take half of G of X and do the composition. I plug X into the GI function and I yield for X Plus five and the F function says Take one divided by the input one divided by and this is my input four x plus five and lo and behold, boom! It works. That's a that's G okay,

So I want a ancient acts to be two X minus five to the third power. But I want h of X to be the composition of two functions. I want f composed of G of X, which is app off g of X to end up being equivalent to H of X. And I wanna find out what are those two functions? What's alphabets and what's g of X? And if I look at it compared this to my original, I could kind of seen it more. See it more plainly. I can see that my inner function, which is right here, is gonna have to be that two X minus five. And I want the outer function f to take whatever the input is and cubit. So that is what f and g of X would have to be. And I could check it. I could say, OK, let me try that. I do that composition. This would say plug X into the GI function, and that would increase my number by a factor of two and then take away five and then the F function takes the input and cubes it, and this is what I would want to be cute and I get exactly what I want. So those air my f on my g I can't switch them.

Okay, so we want to find a chair backs such That's where we found H of X. And we want to find F composed of gene, which is H of X. So that's F of G of X. So we can start by setting G of X to be what's in order. Nominee that's X minus five and then a flex would be equal to one over x.


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