Consider the function f(x) = sqrt(16 - x^2) / (x^2 - 1). What is the domain of f?
Show the answer and explanation
Correct answer
A. [-4, -1) ∪ (-1, 1) ∪ (1, 4]
Principle or equation
The domain consists of all real numbers x for which the expression is defined: the radicand must be nonnegative, and the denominator must not be zero.
Why this answer is correct
For sqrt(16 - x^2), require 16 - x^2 ≥ 0, so x ∈ [-4, 4]. Denominator x^2 - 1 ≠ 0, so x ≠ ±1. Intersecting gives [-4, -1) ∪ (-1, 1) ∪ (1, 4].
Example
For f(x) = sqrt(9 - x^2)/(x^2 - 4), domain is [-3, -2) ∪ (-2, 2) ∪ (2, 3].
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