Consider the function f(x) = sqrt(3x + 6) / (x^2 - 2x - 3). What is the domain of f?
Show the answer and explanation
Correct answer
A. [-2, -1) ∪ (-1, 3) ∪ (3, ∞)
Principle or equation
The domain of a function involving a square root and a rational expression requires the radicand to be nonnegative and the denominator to be nonzero.
Why this answer is correct
For sqrt(3x + 6) to be defined, 3x + 6 ≥ 0, so x ≥ -2. The denominator x^2 - 2x - 3 = (x - 3)(x + 1) must not be zero, so x ≠ 3 and x ≠ -1. Combining these gives x ∈ [-2, -1) ∪ (-1, 3) ∪ (3, ∞).
Example
For f(x) = sqrt(x - 1)/(x - 2), the domain is x ≥ 1 and x ≠ 2, i.e., [1, 2) ∪ (2, ∞).
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