Final answer:
The function h(x) can be expressed as f(g(x)) by choosing f(x) = 2√x + 8 (option C), where g(x) = (x + 4), resulting in the composition matching the original h(x) = 2x + 8.
Step-by-step explanation:
The function h(x) = 2x + 8 can be decomposed into a composition of two functions such that h(x) can be expressed as f(g(x)), where g(x) = (x + 4). The inner function g(x) clearly adds 4 to the input. Now, in order to find the correct outer function f(x), we need to adjust g(x) to become (x + 8), which implies we require another operation to effectively double the input. Examining the options provided, we identify that option D, f(x) = x² + 8, after substituting g(x) into f, would not yield the function h(x) as it lacks the correct coefficients. Option B, f(x) = x² + 16, also can't be the correct function since substituting g(x) into f would lead to a quadratic expression, not the linear h(x). Similarly, option A, f(x) = √x + 8, is incorrect because it involves a square root, changing the nature of the function entirely. The only option that allows for h(x) to be written correctly after composition is option C, f(x) = 2√x + 8, as the square root will undo the square in g(x), so f(g(x)) effectively becomes 2(x + 4) + 8, which simplifies to 2x + 16 - 8, resulting in the original h(x) = 2x + 8. Therefore, the function f(x) that satisfies the composition f(g(x)) = h(x) is 2√x + 8.