find functions of f and g so that fog = H
h(x)=(3x+2)^6
please show work
Hint:
Can you find g if f(x)=x^6 ?
To find functions f and g such that fog = H, we need to determine the composition of functions f and g and simplify it to match the given function H(x) = (3x + 2)^6.
Let's break down the composition of functions fog step by step:
1. Start with the composition of f and g:
fog(x) = f(g(x))
2. Assume that g(x) = 3x + 2, so g maps x to 3x + 2.
3. Now, we need to find a function f such that f(g(x)) = H(x).
Since g(x) = 3x + 2, we have f(3x + 2) = H(x).
4. To simplify the expression and find f, let's substitute the g(x) value into the f(3x + 2) equation:
f(3x + 2) = (3(3x + 2) + 2)^6
5. Simplify the expression inside f:
f(3x + 2) = (9x + 6 + 2)^6
= (9x + 8)^6
Therefore, the desired functions f and g are:
f(x) = x^6, and
g(x) = 3x + 2
as f(g(x)) = (9x + 8)^6 = H(x).
To find the functions f and g such that fog = H, we need to determine the composition of functions fog and the given function H.
First, let's rewrite h(x) = (3x + 2)^6 in terms of fog.
fog(x) = h(x)
fog(x) = (3x + 2)^6
Now, let's break down the composition of functions fog = f(g(x)).
g(x) = 3x + 2
To find f(x), we need to substitute g(x) into h(x):
f(x) = h(g(x))
f(x) = (3(3x + 2) + 2)^6
f(x) = (9x + 6 + 2)^6
f(x) = (9x + 8)^6
Therefore, the functions f(x) = (9x + 8)^6 and g(x) = 3x + 2 satisfy fog = H.