Final answer:
The domain of a function consists of all the possible input values, while the range consists of possible output values. Random variables can have numerical or non-numerical domains. The domain and range help guide proper function application to avoid miscalculations.
Step-by-step explanation:
The domain and range of a function are concepts that refer to the sets of possible input (x-values) and output (y-values) for the function, respectively. For example, considering a function f(x) defined for all real numbers between 0 and 20, the domain of f(x) would be [0, 20]. If the output of this function is the number of books checked out by students from a university library, the domain might include all integers from zero to the maximum number of books a student is allowed to check out.
Random variables such as X, Y, and Z can have domains that are either numerical or non-numerical. For instance, X might have a domain consisting of different student majors, whereas Z will include any non-negative monetary amount representing the expenditure on books. When values appear outside the anticipated range, such as a negative expenditure on books, it suggests an error or an outlier, as in the case where z = -7, as spending cannot be negative.
The domain of a random variable is crucial because it informs us of the potential values that the variable can take before an experiment or observation. Similarly, we decide the range by identifying all possible outcomes or results of the function. Understanding the domain and range helps prevent miscalculations and ensures the proper application of mathematical concepts like probability distribution functions for discrete random variables.