In mathematical terms, how is a function defined?

A function is defined as a relation that assigns exactly one output for each input. This means that for every input value (or element in the domain), there is one and only one corresponding output value (or element in the codomain). This property is crucial because it ensures that the relationship between inputs and outputs is consistent and predictable.

For instance, if you consider a simple function like f(x) = x + 2, for every value of x that you choose, you can determine a single output by applying the function rule. If the definition of a function were to allow multiple outputs for a single input, entering the same input would yield different results, which contradicts the very definition of a function.

The other options describe scenarios that do not fit the definition of a function. They imply either multiple outputs for the same input, no output, or arbitrary outputs, which each violate the primary characteristic of a function. Thus, the notion that a function must provide exactly one output per input is fundamental to its definition in mathematics and helps maintain clarity and usability in mathematical analysis and computations.