Consider the function f(x) = sqrt(2 - x) / (x^2 - 9). Which of the following is the domain of f?
Show the answer and explanation
Correct answer
C. (-∞, 2] except 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 fraction, the denominator cannot be zero.
Why this answer is correct
The square root requires 2 - x ≥ 0, so x ≤ 2. The denominator x^2 - 9 = (x - 3)(x + 3) cannot be zero, so x ≠ 3 and x ≠ -3. Since 3 > 2, it is already excluded by the square root condition. Thus the domain is (-∞, 2] except x = -3.
Example
For f(x) = sqrt(x) / (x - 1), domain is [0, ∞) except x = 1.
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