Question

Find​ f(x) and​ g(x) such that ​h(x)equals​(fcircle​g)(x). h left parenthesis x right parenthesis equals left parenthesis 2 x plus 4 right parenthesis Superscript 6

Answer 1

• f(x) = x^3
• g(x) = (2x+4)^2

Explanation:

There are many ways to decompose h(x) into f(x) and g(x). The main purpose of the exercise seems to be to get you to think about the operations that are performed on x, then divide that list of operations into two parts.

In the function ...

h(x) = (2x +4)^6

the variable x is ...

• multiplied by 2
• 4 is added to the sum
• the sum is raised to the 6th power

Of course, the 6th power can be considered as the cube of a square or the square of a cube, if you like.

In the decomposition shown in the answer above, we have chosen to put most of this list in g(x), including the square of the sum. Then we have made f(x) be the cube of that square.

h(x) = f(g(x)) = (2x+4)^6

When f(x) = x^3, this is ...

h(x) = f(g(x)) = g(x)^3 = ((2x+4)^2)^3

so ...

g(x) = (2x+4)^2 . . . and . . . f(x) = x^3

_____

Other possible decompositions are ...

• g(x) = 2x
• f(x) = (x+4)^6

or

• g(x) = (x+2)
• f(x) = (2x)^6

or

• g(x) = 2x+4
• f(x) = x^6

or ... (many others)