What is the difference between what is and isn't a function?

Answer 1

A function is, mathematically speaking, an object that takes elements from one set and sends them to another set. For example, I can define a function from the set {1, 2, 3, 4} to the set {A, B, C, D} so that 1 is sent to A, 2 is sent to B, 3 is sent to C, and 4 is sent to D. I could also send 1, 2, 3, and 4 to A, leaving the other three letters untouched.

A more common way to define functions is by equations. For example, f(x) = x+1. For any value x, there's an associated value for f(x).

The only relevant property of a function is that for every element in its domain, it must be well-defined, namely the function value must be unique. If f sends the value '2' to both 5 and 7, then f is not a function, because the value at '2' is not well-defined.