Answer:
(b) f(x) = -1/x⁹ and g(x) = -8x +4
(d) f(x) = x⁹ and g(x) = -1/(-8x +4) . . . . . alternate solution
Explanation:
You want to decompose h(x) = -1/(-8x +4)⁹ into f(x) and g(x) such that h(x) = f(g(x)).
Composition
The composition h(x) = f(g(x)) means that the function g(x) will replace x in the definition of f(x).
It is often convenient to look at the order of operations when asked to decompose a function like this. Here, the parenthetical expression (-8x+4) is raised to the 9th power and its opposite reciprocal is found. This suggests that f(x) can be a function that finds the opposite reciprocal of a 9th power, matching choice B. Thus, a reasonable choice is ...
(b) f(x) = -1/x⁹ and g(x) = -8x +4
Also ...
We note that the reciprocal of a 9th power is also the 9th power of a reciprocal. A negative sign is preserved by the odd power. This means that another reasonable choice for the decomposition is ...
(d) f(x) = x⁹ and g(x) = -1/(-8x +4)
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Additional comment
We list choice B first because that one is probably the one you're supposed to claim as the answer. However, this question has two correct decompositions among those listed. You may want to discuss this with your teacher.
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