Finding a Domain from an Equation

Finding a domain from an equation means excluding what breaks the rule — zero denominators out, negative radicands out under even roots — and writing the survivors in interval notation.

Find the domain of f(x) = 1/(x − 3). One hazard: the denominator zeroes at x = 3. Domain: (−∞, 3) ∪ (3, ∞). Now g(x) = √(x − 2). One hazard: an even root refuses negatives, so x − 2 ≥ 0, meaning x ≥ 2. Domain: [2, ∞) — bracket at 2, because zero under the radical is legal. A rule with both hazards runs both hunts and keeps the intersection. Most formulas have no hazards at all; their domain is every real number, written (−∞, ∞).