What is the domain of the function f(x) = sqrt(3 - x) / (x^2 - 5x + 6)?
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Correct answer
A. (-∞, 3] except x = 2 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. For a rational expression, the denominator cannot be zero.
Why this answer is correct
The square root requires 3 - x ≥ 0, so x ≤ 3. The denominator factors as (x - 2)(x - 3), so x cannot be 2 or 3. Since x = 3 is already excluded by the square root condition, the domain is (-∞, 3] with x ≠ 2 and x ≠ 3, which is (-∞, 3] except x = 2 and x = 3.
Example
For f(x) = sqrt(x - 1)/(x - 2), domain is [1, 2) ∪ (2, ∞).
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