f(x)= x/(1+x) g(x) = sin2x Find the following (along with their domains) a) fog b) gof c)fof d) gog

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f(x)= x/(1+x) g(x) = sin2x Find the following (along with their domains) a) fog b) gof c)fof d) gog

Mathematics
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@jim_thompson5910 help please!
a) fog f(g(x)) = f(sin2x)=sin2(1/(1+x))
Hmm that fog looks a little mixed up :d

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\[\large\rm f(\color{orangered}{x})=\frac{\color{orangered}{x}}{1+\color{orangered}{x}}\]Becomes:\[\large\rm f(\color{orangered}{g(x)})=\frac{\color{orangered}{\sin2x}}{1+\color{orangered}{\sin2x}}\]Ya? :o
What you posted: \(\large\rm sin\left(2\color{orangered}{\frac{1}{1+x}}\right)\) This is actually the function f being plugged into g, \[\large\rm \sin\left(2\color{orangered}{\frac{1}{1+x}}\right)=\sin\left(2\color{orangered}{f(x)}\right)=g(\color{orangered}{f(x)})\]Where,\[\large\rm \sin(2\color{orangered}{x})=g(\color{orangered}{x})\]
Just a second, trying to work it out on paper :)
haha yeah I wrote it correctly on paper but did it wrong on here xD sorry! just a moment.
:3
For part A, do you understand how to find the domain of the composition?
I am only having issues with the domain for c now. The rest are all entered and correct.
c) x/1+2x Domain: (-infinity, -1/2) union (-1/2, infinity) it keeps saying the domain is wrong?
Oh sorry I ran off for a sec >.<
You're all good :)
Hmm, I forget how these compositions work sometimes. I guess we need to carry this information around with us before simplifying,\[\large\rm f(x)=\frac{x}{1+x},\qquad\qquad x\ne-1\]So when we get to this composition we have,\[\large\rm f(f(x))=\frac{x}{1+2x},\qquad\qquad x\ne-1,-\frac{1}{2}\]Try that maybe? :o
aaaah. so in interval notation it would be: (-infinity, -1) union (-1,-1/2)union(-1/2, infinity)
Mmmmm yah I think so :)
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Yaay! That is correct. Finally! Thank you so much :D I hadn't thought of that. So it is important to keep in mind the domains for the original functions
Yah, that's kinda tricky! :)
You and your gorgeous drawings xP thanks. That makes sense.
XD
need moar light bulbs ...... to bury
hahah we'll see.
domain of nested functions like that is the intersection of each domain, for f(g(x)) , domain of g and domain of f of g.

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