What Is a Function?

A function is a rule that takes an input and produces exactly one output. We write $y=f(x)$, read “y equals f of x”: feed the function a number $x$, and it hands back a single number $y$.

The set of every input a function is allowed to accept is its domain. The set of every output it can produce is its range. A function can be defined by only part of the number line — $\sqrt{x}$ refuses negative inputs, and $1/x$ refuses zero, because neither has a well-defined real output there.

The one non-negotiable rule: each input maps to exactly one output. A relation that sends a single input to two different outputs (like “the square roots of x”, which gives both $+2$ and $-2$ for $x=4$) is not a function — that’s why we define $\sqrt{x}$ to mean only the non-negative root.