What defines a function in mathematics?

A function in mathematics is defined as a relation that uniquely associates each element of a set with exactly one element of another set. This means that for every input value in the domain (the first set), there is a single, distinct output value in the codomain (the second set). The key aspect of this definition is the uniqueness: no input can produce more than one output, which makes it possible to predict the output based solely on the input.

For example, consider the function f(x) = x^2. For each input value of x, say 2, there is a unique output of 4. If we were to allow multiple outputs for a single input, such as pairing 2 with both 4 and 5, we would no longer have a function under this definition.

This definition not only provides clarity on how functions operate but also establishes the fundamental concept that every input must correspond to one and only one output, ensuring consistent and predictable behavior, which is vital for further mathematical exploration and applications.