Consider the following function definition.Domain: The set of months {January, February, ..., December}.Codomain: All integers in the range [1, 31].Rule: f(month) = number of days in that month in 2025Which of these statements is true with respect to this function?

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Firstly, recall the setup: the domain is the 12 months, the codomain is the set of integers from 1 to 31, and f(month) returns the number of days in that month for the year 2025. Since 2025 is not a leap year, February has 28 days. January, March, May, July, August, October, and December each have 31 days, while April, June, September, and November have 30 days. This means the image of f is the set {28, 30, 31}.

Option a. 'The image is a proper subset of the codomain.' This statement is true because every value in the image (28, 30, 31) lies within the codomain {1, 2, ..., 31}, so the image is a subset of the codomain. Moreover, the image does not contain every element of the codomain; for example, 1, 2, ..., 27, 29, and others are not attained by f. Therefore, it is not equal to the entire codomain, which makes it a proper subset.

To see why the image is not the whole codomain, notice that the codomain includes many integers that never occur as the number of days in any month (such as 1 or 2 days, or 7 days, etc.), and f(month) only yields 28, 30, or 31 for this particular year. Hence, the claim that the image is a proper subset is accurate.

If we were to evaluate other possibilities (even though they aren’t listed), we would find statements like 'the image equals the codomain' would be false since 1–27 and 29–31 are not all attained, and 'the image is not a subset' would be false since all image values are contained in the codomain. The core idea is: the image sits inside the codomain, but does not cover it completely, confirming the proper-subset relationship.

Consider the following function definition.Domain: The set of months {January, February, ..., December}.Codomain: All integers in the range [1, 31].Rule: f(month) = number of days in that month in 2025Which of these statements is true with respect to this function?

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