Final answer:
A piecewise relation defines a function if each input within its domain correlates to exactly one output. A horizontal line within a specific domain, such as f(x) = 5 for 0 ≤ x ≤ 20, is an example of a function since it aligns with the function's definition and passes the Vertical Line Test.
Step-by-step explanation:
To determine which piecewise relation defines a function, we need to consider the Vertical Line Test and the definition of a function. A function describes a relationship where each input (x-value) has exactly one output (y-value). An example of such a piecewise function can be articulated when considering economic models, where the function represents a clear relationship or definition, such as f(x) = a constant value within a specific domain.
In the context of a horizontal line, if we have the function f(x) that is defined as a horizontal line within the domain 0 ≤ x ≤ 20, it represents a function because for any x-value within the domain, there is exactly one y-value. This satisfies the definition of a function. For example, if our function is f(x) = 5 for all x in the domain 0 ≤ x ≤ 20, each x-value is associated with the single output y-value of 5, which complies with the definition of a function.
When we discuss mathematical functions in the realm of probability, such as continuous probability functions, we are often referring to a probability density function that behaves as a function. Thus, for a piecewise relation to define a function, it must pass the specific criteria that every x-value within its domain has exactly one corresponding y-value, which could be visualized graphically as a horizontal line within certain bounds, ensuring that it does not fail the Vertical Line Test.