Final answer:
The value of x where f(x) = 0 is the one that lies between -5 and -1, as the function is continuous and changes sign between these two points. Since choices x = -5 and x = -1 are given values already, and x = 10 is out of the interval, the plausible answer is x = 0.
Step-by-step explanation:
If f(x) is a continuous function defined for all real numbers, and we know that f(-1) = 1, f(-5) = -10, and f(x) = 0 for one and only one value of x, we can determine which of the provided options could be that x value. Since continuous functions have no breaks, jumps, or holes in their graphs, we can apply the Intermediate Value Theorem, which states that if a continuous function has values of opposite signs at two points, then it must cross the x-axis (i.e., take a value of zero) at some point between those two points.
Given that f(-1) = 1 and f(-5) = -10, it's clear that the function changes from positive to negative (or vice versa) between -5 and -1. Therefore, the x-value where f(x) = 0 must lie between -5 and -1. Among the options provided, x = -5 and x = -1 are the values at which we already know the function does not equal zero. Option c) x = 0 falls within that interval and is a plausible answer since no other x-values are given that lie between -1 and -5. Choices x = 10 does not fall within the interval where the function must cross the x-axis based on the information given, therefore it can be eliminated.