SL 2.2—Functions, notation domain, range and inverse as reflection

A function is a mathematical rule that takes each input value and produces exactly one output value. Think of it like a machine: you feed in a number $x$, and the machine returns a single result $f(x)$.

Function: A function $f$ is a rule that assigns to each element $x$ in the domain exactly one element $f(x)$ in the range.

The critical requirement is the one output per input rule. If a single input could produce two different outputs , for example, $f(x)=\pm \sqrt{x}$ gives both $+2$ and $-2$ when $x=4$ , then the relation is not a function.

The standard way to write a function is $f(x)$, where:

• $f$ is the name of the function
• $x$ is the input variable
• $f(x)$ is the output

Other letters are used when the context calls for it:

• $v(t)$ , velocity as a function of time
• $C(n)$ , cost as a function of number of items
• $h(t)$ , height as a function of time

You may also see the mapping notation:
$f:x\to y$
This is completely equivalent to writing $f(x)=y$. It simply says "the function $f$ maps input $x$ to output $y$."

Evaluating a function at a specific value means substituting that value in place of $x$: