What defines a function in mathematics?

A function in mathematics is defined as a relation that assigns exactly one output (y-coordinate) for each input (x-coordinate). This means that for every value of x, there can be only one corresponding value of y. The option stating that a function is a relation in which different ordered pairs have different first coordinates captures this concept accurately by emphasizing that no two ordered pairs can share the same first coordinate if they yield different outputs.

In contrast, other options do not adhere to the definition of a function. The second option allows for the possibility of repeated x-coordinates, which could lead to multiple y-values for a single x-value, thereby violating the definition of a function. The third option suggests that any two ordered pairs can share the same first coordinate, which again compromises the uniqueness of y-values for each x-value. The fourth option references ordered pairs with identical y-coordinates, which does not address the requirement for the x-coordinates and can exist in both functional and non-functional relationships.

Thus, the defining property of a function is indeed that it must provide a unique output for each input, as effectively outlined in the correct choice.