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 Composite function help

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I've been away from school recently due to a nose operation and this monster has appeared in my homework. I understand when there are two equations included, (usually g(x) and h(x). But to find the third, and from what they equal to. If someone could walk me through this I'd heavily appreciate it

 

f(x) = g(h(k(x))) = sin^2(6x) , determine the functions g(x), h(x) and k(x)

 Mar 21, 2017

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 #1
avatar+33615 
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One possibility is as follows:

 

\(k(x)=x^{1/2}\\h(x)=6x^2\\g(x)=\sin^2 x\)

 

Note that \(h(k(x))=6(x^{1/2})^2\rightarrow6x\)

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 Mar 21, 2017
edited by Alan  Mar 21, 2017
 #2
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How'd you attain these posibilities?? Thanks heaps by the way.

Guest Mar 21, 2017
 #3
avatar+33615 
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I started by breaking 6x into 6(x1/2)2 and thinking of x1/2 as k, so that h is then 6k2, which leaves g as sin2h.

 

It is not a unique breakdown! For example we could also have k = (6x)1/3, h = x^3, g = sin2x

or an infinite number of other possibilities!

.

Alan  Mar 21, 2017

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