You are auditing a newly implemented HR database that links employee ID numbers ($x$-values) to their assigned department codes ($y$-values) using ordered pairs. During the audit, you review the following data set: $\{(105, 12), (108, 14), (105, 16), (110, 12)\}$. Recalling the definition of mathematical functions, what is the specific reason this set of ordered pairs does NOT represent a function?

A manufacturing supervisor records part data in a quality control log where the Part ID is the input ($x$) and the weight in grams is the output ($y$). The log contains the ordered pairs ${}(702, 45.3)$ and ${}(702, 45.8)$. True or False: This relation represents a mathematical function.

Auditing a Human Resources Database

As a Quality Control Auditor for a manufacturing database, you are verifying if the Part IDs ($x$) and their measured Tolerance Deviations ($y$) represent a mathematical function. Use your recall of function definitions to match the following terms with the correct descriptions based on this specific audit log: $\{(9, -3), (4, -2), (1, -1), (0, 0), (1, 1), (4, 2), (9, 3)\}$.

A financial analyst is reviewing a set of price changes ($y$) relative to a baseline index ($x$), recorded as the set of ordered pairs: $\{(9, -3), (4, -2), (1, -1), (0, 0), (1, 1), (4, 2), (9, 3)\}$. According to the definition of a mathematical function, this data set is not a function because the index value 9 is associated with two different price changes (-3 and 3), and the index value 4 is associated with both -2 and ____.