Final answer:
The domain is the set of possible input values, and the range is the set of possible output values for a function, while a histogram represents data distribution without a 'domain' and 'range' in a mathematical function sense. When constructing a histogram, one should divide the data into intervals and scale the axes appropriately. For random variables, the domain could be categorical or numerical, while the range of a discrete numerical variable is the set of values it could take.
Step-by-step explanation:
The domain and range of a function are important concepts in mathematics. The domain represents all the possible input values for the function, while the range represents all possible output values. When it comes to functions that are represented graphically, such as histograms, it's crucial to accurately define these sets. However, the question as posed seems to have a typo or an irrelevant part, asking for a domain and range in the context of constructing a histogram, which typically does not have a 'domain' and 'range' like a function does. Instead, histograms represent the distribution of a dataset within specified intervals.
In constructing a histogram, one would typically divide the data into five to six intervals, sketch the graph with a ruler and pencil, and scale the axes to appropriately represent the data. Scaling involves labeling both the horizontal axis (usually the intervals or categories) and the vertical axis (usually the frequency or count of data points within each interval) in a way that accurately reflects the distribution of the data.
When dealing with a random variable (RV), its domain is the set of possible values it can take on. If the random variable is categorical, like hair color, its domain could be expressed in words like {black, blond, gray, green, orange}. Unlike with numerical data, categorical data does not have a range in the same sense, but rather a set of categories.
When dealing with numerical data, such as the number of classes taken or the amount of money spent on books in the last semester, for discrete numerical data, the domain would be a set of integers, and the range would be the set of values those variables could take. For continuous numerical data, the range is typically given in interval notation.
The essential characteristics of a discrete probability distribution include having a finite or countably infinite set of values and associated probabilities that sum to one. A value such as z = -7 for the amount of money spent on books is not possible, assuming you cannot spend negative money on books.