What defines a function in mathematics?

A function is fundamentally defined as a relation that assigns exactly one output for each input. This characteristic ensures that for every input value in the function's domain, there is a single, unique output in the codomain. This unique pairing is essential for the definition of a function, as it allows the function to be predictable and consistent in its behavior.

For instance, if we consider the function f(x) = x^2, each input value of x corresponds to one and only one output value of f(x). This rule enables us to determine the output whenever we know the input. If a relation were to assign multiple outputs for a single input, it would not be classified as a function, as this would violate the core principle of uniqueness that defines functional relationships.

Thus, the correct understanding of a function hinges on this one-to-one relationship between inputs and outputs, solidifying the definition provided in the chosen answer.