Final answer:
The domain of a function is the set of all possible input values for which the function is defined, and it can be expressed using interval notation, set builder notation, or words. It is important for understanding and interpreting functions in mathematical and real-world contexts.
Step-by-step explanation:
To write the mathematical domain of a given function, you need to identify the set of all possible input values (x-values) for which the function is defined. The domain can be expressed in several ways, including interval notation, set builder notation, or verbally describing the set of values.
For example, consider the function f(x) for 0 ≤ x ≤ 20, with x being a real number. If the graph of f(x) is a horizontal line, the function has a constant value for all x within the domain. However, since the function is defined only for values of x between 0 and 20, inclusive, the domain of f(x) can be written as [0, 20] in interval notation, indicating that x can take any real number value within this closed interval.
In the case of random variables (RVs), the domain does not have to be numerical. For instance, if X = hair color, the domain would be expressed in words as the set {black, blond, gray, green, orange}. We determine the specific value of a random variable only after observing the outcome of an experiment.
Understanding domain is crucial for correctly interpreting functions and applying them to real-world situations, such as determining the number of books checked out from a library or the range of possible values for discrete probability distributions.