Consider the function f(x) = sqrt(2x - 6) / (x^2 - 16). What is the domain of f?
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Correct answer
A. [3, 4) ∪ (4, ∞)
Principle or equation
The domain of a function is the set of all real numbers for which the function 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 radicand 2x - 6 must be ≥ 0, so x ≥ 3. The denominator x^2 - 16 = (x - 4)(x + 4) must not be zero, so x ≠ 4 and x ≠ -4. Since -4 is not in [3, ∞), we only exclude 4. Thus the domain is [3, 4) ∪ (4, ∞).
Example
For f(x) = sqrt(x - 2) / (x^2 - 9), the domain is [2, 3) ∪ (3, ∞) because x ≥ 2 and x ≠ 3, -3 (but -3 is not in [2, ∞)).
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