Range

On a graph

The range is how far the graph spreads up and down — which $y$-values it reaches.

Sweep the graph from bottom to top and note the heights it occupies, the same way you sweep left to right for the domain.

Range does not copy the domain

This is worth checking separately every time.

Function	Domain	Range
$f(x) = 2x + 1$	all reals	all reals
$f(x) = 5$	all reals	just $\{5\}$
$f(x) = \sqrt{x}$	$x \ge 0$	$y \ge 0$
$f(x) = x^2$	all reals	$y \ge 0$

A horizontal line accepts every input but produces only one output. A parabola accepts every input but never returns a negative, because a square cannot be negative.

As a set

A range is a set of values, so a repeated output is written once. If three different inputs all give $7$, then $7$ appears in the range a single time.

Which is which

Domain before range, $x$ before $y$ — the two orders match, which is a reliable way to keep the labels straight under pressure.