Question

What are two major concerns for the domain of a function?

a) Range and continuity
b) Endpoint behavior and symmetry
c) Monotonicity and concavity
d) Differentiability and integrability

Answer 1

Option (c), The two major concerns for the domain of a function are the function's continuity (all values are defined) and its single-valued nature (each input corresponds to one output). Examples include domains for student majors, number of classes taken, and money spent on books.

Step-by-step explanation:

Two major concerns for the domain of a function are whether the function has any points at which it is not defined (discontinuities) and whether the function assumes more than one value for any point in the domain (whether the function is single-valued or multi-valued).

For example, consider these scenarios:

1. If X represents a student's major, the domain of X would be all possible majors offered by an educational institution. However, a student cannot have two majors simultaneously in this context, so the function must be single-valued. Also, every student must have a major, so there's no discontinuity in the domain.
2. If Y is the number of classes taken in the previous semester, the domain must include all non-negative integers, as one cannot take a negative number of classes—this ensures that the function is defined over its entire domain.
3. If Z is the amount of money spent on books in the previous semester, then the domain would likely include all non-negative real numbers (since you cannot spend a negative amount of money), ensuring the function's continuity over its domain.

Any values outside of these conditions would not be in the domain of these functions, such as a student having two majors at the same time (X), a negative number of classes (Y), or spending a negative amount of money (Z).