Problem

Find the complete real domain of \(\sqrt{x^2~-~9}\). State the factorization, radicand condition, endpoint inclusion, and intervals.

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For a real square root, analyze the original radicand before doing any simplification. Factoring identifies where its sign can change, and the product is nonnegative where its factors share a sign. Because a radicand of zero is allowed, any boundary that makes the product zero is included rather than discarded.

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Factorization: (x-3)(x+3). Condition: both factors ≥0. Endpoints: 3 included; -3 excluded. Domain: [3,∞).

Factorization: (x-3)(x+3). Condition: product ≤0. Endpoints: -3 and 3 included. Domain: [-3,3].

Factorization: (x-3)(x+3). Condition: product >0. Endpoints: -3 and 3 excluded. Domain: (-∞,-3)∪(3,∞).

Factorization: (x-3)(x+3). Condition: product ≥0. Endpoints: -3 and 3 included. Domain: (-∞,-3]∪[3,∞).

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Find the complete real domain of \(\sqrt{x^2~-~9}\). State the factorization, radicand condition, endpoint inclusion, and intervals.

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