Final answer:
The domain of a function is the set of all permissible inputs, while the range is the set of all potential outputs. Variables X, Y, and Z from the examples are random variables that each have a specific domain. The range of a function like f(x) in the example, which is a horizontal line, is the value of the line within the domain of observed x values.
Step-by-step explanation:
Finding the Domain and Range of Functions
The domain of a function refers to the set of all possible inputs (x-values) for which the function is defined. Conversely, the range is the set of all possible outputs (y-values) that the function can produce. Considering the given information, let's discuss the domain and range of the functions represented by the random variables X, Y, and Z separately.
The domain of X, which represents a student's major, is the list of all majors offered at the university. It may include majors such as English and Mathematics, along with an option for being undeclared.
The domain of Y, representing the number of classes taken in the previous semester, includes all non-negative integers up to the maximum allowed by the university.
The domain of Z, reflecting the amount of money spent on books in the previous semester, includes all non-negative monetary amounts, starting from zero.
It's important to note that X, Y, and Z are considered random variables because they can assume different values within their domain based on the outcomes of an experiment or real-world observations. For example, the specific value of X can only be determined after surveying a student's major.
In terms of the range, it is defined by the behavior and constraints of the function. In the given example of a function f(x) being a horizontal line, the function's output is constant; hence its range would be that constant value. However, the reliable range is subjected to the values within the domain of x that was observed.
As for the value z = -7, it is an impossible value for Z since its domain includes only non-negative values. Negative amounts of money spent would not make sense in this context. Lastly, the two essential characteristics of a discrete probability distribution are: that each possible value is assigned a probability between 0 and 1, and that the sum of the probabilities for all possible values is exactly 1.