SOLUTION: Find functions f and g such that h= g o f. (Note: The answer is not unique. Enter your answers as a comma-separated list of functions. Use non-identity functions for f and g.)

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Question 1160675: Find functions f and g such that h= g o f. (Note: The answer is not unique.
Enter your answers as a comma-separated list of functions. Use non-identity
functions for f and g.)
h(x)=(5x-3)^3/2
{g(x), f(x)} =
Found 3 solutions by MathLover1, Edwin McCravy, greenestamps:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

if = o
and
then and

{ , }
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!


g∘f(x) means to take the whole right side of f(x), put it in parentheses and
substitute it for x in the right side of g(x).

So if you put parentheses around 5x-3 and substituted it for x in x3/2 you would get 

(5x-3)3/2 

So make g(x) = x3/2 and f(x) = (5x-3).

The function on the right in g∘f, which is f, is the one you start with.

Since you're stating with 5x-3, make

            f(x) = 5x-3

Then take

            g(x) = x3/2

and then when you take the whole right side of f(x), which is 5x-3, put
parentheses around it, and substitute it for x in the right side of g(x).,
you get:

           g∘f(x) = (5x-3)3/2

So the answers are:
              
            g(x) = x3/2,  f(x) = 5x-3


Edwin
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


Let me expand on the answer from tutor @Edwin, which shows one pair of functions that satisfy the conditions of the problem.

As your post says, the answer is not unique; there are other possible answers for the functions f and g.

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

We want to perform those three operations using two functions. We have two clear choices:

(a) We can combine operations (1) and (2) into a single function:
f(x) multiplies by 5 and subtracts 3 -->

Then g(x) raises its input to the power 3/2:

Then

That is probably the most "reasonable" way to get h(x) as the composition of two functions.

But we have another obvious possibility....

(b) We can leave the first operation alone and combine (2) and (3) into a single function:




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".

That gives us two more ways to get h(x) as the composition of two functions:

(c) ; ;

or

(d) ; ;


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