Final answer:
Given f(x) is continuous and that it transitions from a positive to negative value between x = -1 and x = -5, option D) x = -3 could be the x-value where f(x) equals zero, according to the Intermediate Value Theorem.
Step-by-step explanation:
If f(x) is a continuous function defined for all real numbers, with f(-1) = 1, f(-5) = -10, and given that f(x) = 0 for only one value, to determine which of the provided options for x could be where f(x) equals zero, we must use the Intermediate Value Theorem. This theorem states that for any value between two endpoints on a continuous function, there exists at least one corresponding x-value on that interval. Since f(x) changes from positive at x = -1 to negative at x = -5, and since we are told that f(x) equals zero for precisely one x-value, we need to find an x between -1 and -5 where f(x) can be zero.
The options A) x = -1 and B) x = -5 can be immediately discarded because we already know these are not the zeros of the function (f(-1) = 1 and f(-5) = -10). Option C) x = 0 cannot be the zero since there is no indication that f(x) switches signs again after x = -5. The solution must be D) x = -3, as it is the only value provided that lies between -1 and -5, where f(x) could potentially change from a positive to a negative value (or vice versa), which aligns with the condition that there's exactly one zero of the function.