Domain

On a graph

The domain is how far the graph spreads left to right — which $x$-values it occupies.

A line drawn with arrows on both ends has a domain of all real numbers. A line segment from $(1, 2)$ to $(5, 10)$ has domain $1 \le x \le 5$, because the endpoints stop it.

What gets excluded, and why

Exclusions are forced by arithmetic, not chosen:

Function	Excluded	Reason
$f(x) = \tfrac{1}{x - 3}$	$x = 3$	denominator becomes zero
$f(x) = \sqrt{x - 4}$	$x < 4$	no real square root of a negative
$f(x) = 2x + 1$	nothing	no denominator, no root

To find an exclusion, set the denominator to zero and solve, or set the expression under a root at or above zero and solve.

Restricting a domain on purpose

A real situation often narrows a domain further. A ticket-cost formula $C(x) = 9x$ accepts $x = -3$ quite happily, but nobody buys $-3$ tickets. The sensible domain is the whole numbers.

The equation does not know what it describes. Restricting the domain is how you tell it — see domain and range.