Final answer:
For a function f(x) = 20 restricted between 0 and 20, the domain is [0, 20] and the range is {20}. For its inverse, the domain would be {20} and the range would be [0, 20], but technically such a function does not have an inverse since it is not one-to-one. In a random variable context, domains can be non-numeric, such as hair colors.
Step-by-step explanation:
The domain and range of an inverse function relate to the set of allowable inputs and outputs for that function. The domain of a function is the complete set of possible values of the independent variable. In the case of a horizontal line function, such as f(x) = 20 for 0 ≤ x ≤ 20, the domain in interval notation is [0, 20]. Since the line is horizontal, all the y-values (outputs) are the same. Therefore, the range in interval notation is [20, 20] or simply {20}, indicating that the only output value is 20.
When considering the inverse of this function, we must swap the domain and range. Thus, for the inverse function, the domain would be {20} (since the original range was {20}), and the range would be [0, 20] (since the original domain was [0, 20]). However, since this is not a one-to-one function, it technically does not have an inverse because each input in the domain does not have a unique output in the range.
Regarding random variables, the domain can be numeric or categorical. For instance, if X = hair color, then the domain is a set of words describing possible hair colors - {black, blond, gray, green, orange}. The particular value that the random variable takes can only be known after an observation or experiment is conducted.