Find functions f & g so that fog = H

H(x)=(3x+2)^6

Please show work

g(x) = 3x+2

f(x) = x^6

f(g) = g^6 = (3x+2)^6

To find the functions f and g such that fog = H(x) = (3x + 2)^6, we need to decompose H(x) into composition form. Let's break it down step by step.

Step 1: Identify the inner function g(x)
Let's start by looking at the exponent in H(x). The term (3x + 2) is raised to the power of 6. This suggests that the inner function g(x) is g(x) = (3x + 2).

Step 2: Identify the outer function f(x)
Now that we have the inner function g(x), we need to find the outer function f(x) that will yield the desired result when composed with g(x). To do this, we want to determine what operation or function would recover the original expression (3x + 2)^6.

To find the outer function, we can consider the reverse operation of raising a base to a power. In this case, the reverse operation is taking the 6th root, or the inverse of raising to the power of 6.

Therefore, the outer function f(x) is f(x) = x^(1/6).

Step 3: Confirm the composition
Now that we have identified the inner and outer functions, we can confirm that fog = H(x).

fog(x) = f(g(x)) = f(3x + 2) = (3x + 2)^(1/6) = H(x)

Hence, the functions f(x) = x^(1/6) and g(x) = 3x + 2 satisfy the condition fog = H(x) = (3x + 2)^6.