We introduce a new query primitive called Function Query (FQ). An FQ operates on a set of math functions and retrieves the functions whose output with a given input satisfies a query condition (e.g., being among top-k, within a given range). While FQ finds its natural uses in querying a database of math functions, it can also be applied on a database of discrete values. We show that by interpreting the database as a set of user-defined functions, FQ can retrieve the information like existing analytic queries such as top-k query and scalar product query and even more. Our research addresses the challenges of FQ execution and authentication. The former is how to minimize the computation and storage costs in processing an FQ, whereas the latter, how to verify that the result of an FQ returned by a potentially untrustworthy server is indeed correct. Our solutions are inspired from the observations that 1) the intersections of a set of continuous functions partition their domain into a number of subdomains, and 2) in each of these subdomains, the functions can be sorted based on their output. We prove the correctness of the proposed techniques and evaluate their performance through analysis, prototyping, and experiments using both synthetic and real-world data. In all settings, our techniques exhibit excellent performance. In addition to FQ, our research has developed another query primitive called Improvement Query, which we also include in this dissertation.