The question asks how the domains of the pieces of a piecewise-defined function relate to the entire domain of the function. We are given four options, and we must choose the correct one.

1. Problem Description

The question asks how the domains of the pieces of a piecewise-defined function relate to the entire domain of the function. We are given four options, and we must choose the correct one.

2. Solution Steps

A piecewise-defined function is defined by multiple sub-functions, each applying to a certain interval of the main function's domain. The domains of these sub-functions must cover the entire domain of the piecewise function. If they only made up half the domain, the piecewise function would be undefined for the other half. Also, the domains of the sub-functions should not overlap, as this would mean that the same $x$ value would have multiple function values, violating the definition of a function.

Thus, the correct answer is the one which says the domains make up the entire domain and there is no overlap.

3. Final Answer

C. The intervals that are the domains of the pieces make up the entire domain of the piecewise function with no overlap.