Final answer:
The domain of a rational expression excludes values that make the denominator zero. For the given examples, the domain of X is a set of declared majors, Y is the set of non-negative integers within the university's limits, and Z is any non-negative real number representing money spent.
Step-by-step explanation:
When finding the domain of a rational expression, one must consider certain restrictions that are primarily based on the properties of real numbers and the operations we can perform on them. Particularly with rational expressions, the denominator cannot be zero as division by zero is undefined. Therefore, the domain of a rational expression consists of all real numbers except for those which make the denominator zero.
Considering the student's questions:
- The domain of X (the student's major), would be a set of all declared majors in the university. It is qualitative and not numerical, reflecting different categories such as {English, Mathematics, ...}.
- The domain of Y (the number of classes taken in the previous semester) is a set of non-negative integers that can be taken, starting from zero up to a maximum that is allowed by the university.
- The domain of Z (the amount of money spent on books) would be any non-negative real number, indicating any amount of money starting from zero and upwards.
Variables X, Y, and Z are referred to as random variables because their values are specific outcomes that can only be determined after an 'experiment' or after collecting data from the students, making them unpredictable beforehand.
If a value of z = -7 is found in the collected data, it would not be a valid value for Z, as the amount of money spent cannot be negative.
Discrete probability distributions, like the one described in the example, require two essential characteristics. Every probability must be between 0 and 1, and the sum of all probabilities must equal 1.
When working with rational expressions or variables in general, it is important to:
- Eliminate terms wherever possible to simplify the algebra.
- Check the answer to see if it is reasonable in the context of the problem.