What is the domain of the function f(x) = sqrt(2 - x) / (x^2 - 9)?
Show the answer and explanation
Correct answer
A. (-∞, 2] except x = -3 and x = 3
Principle or equation
The domain of a function is the set of all real numbers for which the expression is defined. For a square root, the radicand must be nonnegative: 2 - x ≥ 0, so x ≤ 2. For a denominator, it cannot be zero: x^2 - 9 ≠ 0, so x ≠ ±3. Since 3 > 2, only -3 is in the interval (-∞, 2] and must be excluded.
Why this answer is correct
From 2 - x ≥ 0, we get x ≤ 2. From x^2 - 9 ≠ 0, we get x ≠ 3 and x ≠ -3. Since 3 is not ≤ 2, it is already excluded; we must exclude -3. Thus domain is (-∞, 2] except x = -3.
Example
For f(x) = sqrt(1 - x) / (x - 2), domain is x ≤ 1 and x ≠ 2, which simplifies to (-∞, 1].
This published item includes a stored explanation and passed the platform’s publication workflow. It is independent preparation material, not a claim of an official or recalled examination question.