Domain of a Function
The domain of a function is the complete set of all possible input values for which the function is defined.
Definition
The domain is the set of all allowed inputs for a function, every $x$-value you're permitted to plug in so the function produces a real, defined output. Common restrictions: denominators cannot be zero, and expressions under even radicals must be non-negative. Formally, for a function $f: A \to B$, the domain is the set $A$; in analysis, the natural domain of a real-valued function is the maximal subset of $\mathbb{R}$ on which the expression is well-defined, and for complex functions domains extend to subsets of $\mathbb{C}$, where analytic continuation can extend them further.
Example
Think of the domain as the guest list for a party: only those on the list can get in and receive an output. For the function "square root of $x$," you can only plug in $0$ or positive numbers, so the domain is all numbers greater than or equal to $0$. For $f(x) = 1/(x - 3)$, $x$ cannot equal $3$ (division by zero), so the domain is all real numbers except $3$, written $(-\infty, 3) \cup (3, \infty)$; for $g(x) = \sqrt{x + 2}$, we need $x + 2 \ge 0$, so $x \ge -2$ and the domain is $[-2, \infty)$. The gamma function extends the factorial to all complex numbers except non-positive integers, showing how the domain concept applies well beyond elementary functions.
Key Insight
When no domain is stated, assume the natural domain, the largest set of real numbers for which the expression is defined. In topology, the domain carries a topological structure, and continuity, differentiability, and integrability are all properties stated relative to it.