Consider the function f(x) = sqrt(5 - x) / (x^2 - 4). What is the domain of f?
Show the answer and explanation
Correct answer
A. (-∞, -2) ∪ (-2, 2) ∪ (2, 5]
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
The radicand 5 - x must be ≥ 0, so x ≤ 5. The denominator x^2 - 4 must not be zero, so x ≠ ±2. Combining these gives x ≤ 5 with x ≠ -2 and x ≠ 2. In interval notation, that is (-∞, -2) ∪ (-2, 2) ∪ (2, 5].
Example
For f(x) = sqrt(3 - x)/(x^2 - 1), the domain is x ≤ 3 and x ≠ ±1, i.e., (-∞, -1) ∪ (-1, 1) ∪ (1, 3].
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