Which organizational function would directly supervise personnel performing various operations?

The correct answer is line functions because these are the organizational functions responsible for directly supervising personnel who carry out the core operations of an organization. In law enforcement and many organizations, line functions are at the forefront of the operational hierarchy and are crucial for day-to-day activities. They include roles that have either direct authority over subordinates or directly engage with the execution of the organization's primary mission.

In contrast, auxiliary functions typically provide support and resources to the core activities but do not engage directly in operational supervision. Staff functions refer to advisory or support roles that assist line functions but are not involved in the direct command of personnel. Primary functions encompass the main objectives or goals of the organization but do not explicitly denote supervision, which is the critical aspect addressed in this context. Thus, line functions are explicitly focused on command, oversight, and management of personnel involved in operations, making them the best fit for the question asked.