5

Show that $f(g(x))=x$ and $g(f(x))=x$ for all $x:$$f(x)= rac{1}{x+2} ext { and } g(x)= rac{1-2 x}{x}(x eq 0, x eq-2)$...

Question

Show that $f(g(x))=x$ and $g(f(x))=x$ for all $x:$$f(x)= rac{1}{x+2} ext { and } g(x)= rac{1-2 x}{x}(x eq 0, x eq-2)$

Show that $f(g(x))=x$ and $g(f(x))=x$ for all $x:$ $f(x)=\frac{1}{x+2} \text { and } g(x)=\frac{1-2 x}{x}(x \neq 0, x \neq-2)$



Answers

Show that $f(g(x))=x$ and $g(f(x))=x$ for all $x:$ $f(x)=\frac{1}{x+2} \text { and } g(x)=\frac{1-2 x}{x}(x \neq 0, x \neq-2)$

Okay, so in this problem, they want us to its a four part problem. They want us to find the domain of F of G and G F after we compose those two functions together, and then they want to see which conditions make, um, effort G equal to G of f. Okay, so they give us f of X and G of X. Of course, X is an unknown variable, and A and B, C and D are all unknown constant, so they don't change, but they are unknown. So let's do f of G first, and we see that we have, um, f of C X plus d, and that's going to be a Times C X plus D, then add B. When we distribute that we get a c X plus a d waas. He all right, and that's a good start for A for B. We're composing F into G, so that's G of X plus B. And we want this to be plugged in, so that's going to be see Time's a quantity a X Plus B and then add do when we distribute that it should be a C X plus B, C and then at D, Part C asks for the domain of these two composed functions. And luckily, these air both just linear equations and linear equations are polynomial. So the domain is all real numbers. We can write it like that. Or it can say, um, the domain of X is all act such that X is a rial number, all right. And then the last poor is which conditions will make these two things equal to each other. Okay, well, they just have to be equal to each other, so I'm going to set them equal to each other. Um, I have a C X plus a d plus B is equal to a C X plus beast heat plus t. Okay. And when you take a look at this, we have an a C X on both sides. If I were to strip track that from both sides, it would cancel and I'd be left with a D plus B is equal to B C plus D, and unfortunately, I can't go much further than this. So these two things are true when a Times D plus B is equal to B C plus de

All right, We're composing thes two functions together to see that, um, they both went composed together are equivalent to each other and equal X. And what I think this problems trying to do is try to show you a pattern that maybe you recognized in previous problems. I have times a and plus B. So over here I'm going to be dividing by a and subtracting being in order to get in short, these two things, Aaron versus of each other. So let's look at f of G. That's f of one over a times a quantity x minus b some plugging this entire expression in four X here. And I get a times one over a times a quantity x minus b close that parentheses e close it again and then plus B so eight times one over a cancels to just one. So I'm left with X minus B plus B, which is just equal to next. Reversing that process. I have g of a x plus B. So I have one over a times a X plus b that's attract be And, uh, I'm gonna work pen dos. So I want to work inside the innermost parentheses. First I have one over a times a X because B minus B is zero and then one a over sometimes hey is just going to be one necks.

All right, so we're composing one over X with itself. This is an interesting case that they are asking us. Hey, when this happens, you should get acts out. So verify it. We have f of one over X. So that's like 1/1 over x. And, uh, what we see here, this one we can rewrite as 1/1 if we wanted to and see that it's 1/1 times X over one. And that's just a fancy way of saying one Times X, which is X. And since GF is actually gonna be the same thing, we see that the reciprocal function is the one and only function besides the identity of function that you can compose with itself and get acts out. So it's its own inverse.

Miles. Four explosive for and do its ministry minus four minus four. Last. Next, almost two x plus four minus T for the last way. No. 64 Mr. For My next role 171 1.17


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