Final answer:
To find g(f(x)), substitute f(x) into g(x) resulting in g(f(x)) = x^(-6) + 2x^(-3). The domain is all real numbers except x = 0 (x ≠ 0).
Step-by-step explanation:
To find g(f(x)), we first need to plug f(x) into g(x). Given that f(x) = x^(-3) and g(x) = x^2 + 2x, we substitute x in g(x) with x^(-3) to get:
g(f(x)) = (x^(-3))^2 + 2(x^(-3))
g(f(x)) = x^(-6) + 2x^(-3)
The domain of g(f(x)) is all real numbers except x = 0 because at x = 0, the function f(x) would have a denominator of zero, making the function undefined. Therefore, the domain of g(f(x)) is x ≠ 0. We can write the domain as x .