NEED ASAP!! let h(x)= -2/3 sin (3/2x). find 2 functions f and g so that (f o g)(x) = h(x)

Question

h(x) = - 2 sin ( 3 x). find 2 functions f and g so that ( f o g) (x) = h(x)

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3 2

(2 over 3) and (3 over 2)

Answer 1

There are different ways to answer this question.The important thing to remember is that -2/3*sin(3/2x) = (f o g)(x) = f(g(x)). You can basically work from the inside out - g(x) can be the "inside" of h(x), and f(x) can be the rest of it. Exactly where you decide to make the break between "inside" and "outside" doesn't really matter since there aren't any other requirements in the question.

One answer is:

g(x) = 3/2x

f(x) = -2/3*sin(x)

Because now when you do f(g(x)), you're replacing the x in f(x) with the 3/2x from g(x) and so you get -2/3*sin(3/2x) which is h(x).

You could also break it in other places - another solution would be g(x) = sin(3/2x). What would f(x) be in that case? (Hint: What else is in h(x) that isn't in g(x)?)