How did Gottlob Frege distinguish objects from functions?

In the philosophy of logic, Gottlob Frege distinguished between objects and functions based on their logical completeness, a concept he described as "saturation".12

• Objects: Frege characterized objects (or individuals) as "complete" or "saturated" expressions. These represent the entities that serve as the arguments for functions.12
• Functions: In contrast, Frege viewed functions—which include concepts—as "incomplete" or "unsaturated" expressions. He argued that a function contains a "gap" that must be filled by an object to produce a complete value, such as a truth-value.12

Frege’s analogy for this relationship was drawn from mathematics: just as a mathematical function like $\dots \times 4$ requires a number to produce a result, a linguistic predicate like "$\dots$ is a lecturer" requires a name (an object) to form a complete sentence. By isolating these predicate-functions, Frege demonstrated that the truth-value of a sentence is determined by the outputs of these functions when applied to their respective objects.1