Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{Q} $ be defined as $ f(x, y) = \frac{x+2}{y-1} $.

Which of the following best describes $ f $?

(Note: $ A^2 $ is another way of writing $ A \times A $)

A. Not one-to-one, not onto, and not total
B. One-to-one, onto, and not total
C. Total and one-to-one, and not onto
D. Onto and not one-to-one, and not total
E. Total, one-to-one, and onto
F. One-to-one and not onto, and not total
G. Total and not one-to-one, and not onto
H. Total and onto, and not one-to-one

To determine the properties of the function f, we need to consider its injectivity (one-to-one) and surjectivity (onto) as well as the total function.

One-to-one: A function is one-to-one if it maps distinct input pairs to distinct output values. In the given function f(x, y) = (x + 2) / (y - 1), it can be observed that different pairs of (x, y) can yield the same output value. For example, f(1, 2) and f(3, 4) both result in the value of 3/1. Hence, the function is not one-to-one.

A function is onto if every element in the codomain has a preimage in the domain. In this case, the codomain is Q (rational numbers), and it is possible to find certain elements in Q that do not have corresponding preimages in [tex]Z^2[/tex]. For instance, there is no input pair (x, y) that maps to the value 0 in Q. Therefore, the function is not onto.

Total: A total function is defined for every element in the domain. Since the given function is defined for all integer pairs (x, y) except when y = 1 (which results in a division by zero), we can say that the function is total.

Based on the above analysis, the function f is described as one-to-one and not onto.

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