In mathematics, how is a function defined?

A function is defined as a relation in which each input is associated with exactly one output. This means that for every value in the domain (the set of inputs), there is a unique value in the range (the set of outputs). This one-to-one correspondence between inputs and outputs is a fundamental characteristic of functions.

In a function, if you pick any input value, you can determine exactly one output value corresponding to it. This property distinguishes functions from more general relations, where one input might lead to multiple outputs. For example, in the relation where an input represents a person and the output represents their possible lottery numbers, this relation could produce multiple outputs for most of the inputs, which would not satisfy the definition of a function.

Understanding this concept is vital in mathematics since functions serve as the building blocks for more complex mathematical concepts, allowing for predictable behavior and the ability to model real-life situations systematically.