What is the natural domain of the real-valued function f(x) = sqrt((x - 2) / (x + 5))?
Answer and explanation
Correct answer: D. (-infinity, -5) U [2, infinity)
- Step 1: The radicand must be non-negative: (x - 2)/(x + 5) >= 0, with x != -5. Step 2. Testing intervals: for x < -5, ratio is positive. For -5 < x < 2, ratio is negative. For x >= 2, ratio is positive. Step 3. At x = 2, numerator is 0 (included); at x = -5, denominator is 0 (strictly excluded). Answer: (-infinity, -5) U [2, infinity). Why not the others: -5 must be excluded because division by zero is undefined.
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