What defines a function in mathematics?

A function in mathematics is defined as a specific type of relation between two sets where each input (or element from the domain) is associated with exactly one output (or element from the range). This means that for every input, there is a unique output. The concept of ordered pairs is integral to this definition; each input is paired with its corresponding output, maintaining the uniqueness of the output for each distinct input.

The reason that a set of ordered pairs with unique outputs for each input is the correct definition of a function is that it fulfills the essential criteria of single-valuedness—ensuring that no input can map to multiple outputs. This property is crucial for the structure and predictability of functions, as it allows for consistent and reliable mappings between inputs and outputs.

In contrast, other definitions do not fulfill the criteria for being classified as a function. For example, a set with multiple outputs for each input does not meet the unique mapping requirement, and hence cannot be considered a function. A set of unordered pairs does not provide the necessary structure to ensure that each input corresponds only to one output. Lastly, a rule that only includes constants does not encompass the broader definition of functions, which can also involve variable relationships that are not limited to constant values.