Question 1160675
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Let me expand on the answer from tutor @Edwin, which shows one pair of functions that satisfy the conditions of the problem.<br>
As your post says, the answer is not unique; there are other possible answers for the functions f and g.<br>
Consider the sequence of operations the function h(x) does to the input value x:
(1) multiply by 5
(2) subtract 3
(3) raise to the power 3/2<br>
We want to perform those three operations using two functions.  We have two clear choices:<br>
(a) We can combine operations (1) and (2) into a single function:
f(x) multiplies by 5 and subtracts 3 --> {{{f(x) = 5x-3}}}<br>
Then g(x) raises its input to the power 3/2: {{{g(x) = x^(3/2)}}}<br>
Then {{{g(f(x)) = (5x-3)^(3/2)}}}<br>
That is probably the most "reasonable" way to get h(x) as the composition of two functions.<br>
But we have another obvious possibility....<br>
(b) We can leave the first operation alone and combine (2) and (3) into a single function:
{{{f(x) = 5x}}}
{{{g(x) = (x-3)^(3/2)}}}
{{{g(f(x)) = (5x-3)^(3/2)}}}<br>
And in fact there are yet other possibilities.  We could perform the single operation "raise to the power 3/2" as a sequence of two operations -- either "raise to the power 3 and then take the square root" or "raise to the 1/2 power and then raise to the power 3".<br>
That gives us two more ways to get h(x) as the composition of two functions:<br>
(c) {{{f(x) = (5x-3)^3}}}; {{{g(x) = x^(1/2)}}}; {{{g(f(x)) = ((5x-3)^3)^(1/2) = (5x-3)^(3/2)}}}<br>
or<br>
(d) {{{f(x) = (5x-3)^(1/2)}}}; {{{g(x) = x^3}}}; {{{g(f(x)) = ((5x-3)^(1/2))^3 = (5x-3)^(3/2)}}}<br>