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Question 1173920: Given h(x) = cos sqrt(3x^2-7), determine f(x) and g(x) in two different ways, such that h(x) = (f o g)(x) where f(x) is not equal to x or g(x) is not equal to x. For each option, show that h(x) = (f o g)(x).
f(x) = ? g(x) = ?
f(x) = ? g(x) = ?
Answer by greenestamps(13195)   (Show Source):
You can put this solution on YOUR website!
You are to find f(x) and g(x) so that the composition f(g(x)) is equal to
To answer a question like this, consider what you would do to evaluate the expression for a given value of x. In this example, the steps you would take are
(1) square the value
(2) multiply it by 3
(3) subtract 7
(4) take the square root
(5) find the cosine
To write the given function h(x) as the composition of two function f(x) and g(x), you can simply break that string of five operations into two strings.
So one example would be to break the string of operations between (3) and (4). That would give you...
g(x) is operations (1) to (3) -- square the value, multiply it by 3, and subtract 7: 
f(x) is operations (4) and (5) -- take the square root and find the cosine: 
Then 
The problem asked you to find f(x) and g(x) in two different ways; that was one of them.
With five operations in all, there are four places you can break the string in two, so there are three other ways to define f(x) and g(x).
You only need to find one more; however, you will learn more from this if you take the time to find all three remaining ways of defining f(x) and g(x).
Note the way I chose to break the string of operations in two was, quite unintentionally, almost certainly the most logical way. The other options will look much more awkward -- don't let that deter you.
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