Discussion
The domain of the function \(f(x) = \sqrt{x^{2} + 2x - 15}\) is
Answer: D
The domain is \((-\infty, -5] \cup [3, \infty)\).
The square root needs a non-negative argument:
\[ x^{2} + 2x - 15 = (x + 5)(x - 3) \ge 0 \]
The product is non-negative outside the roots and zero at them, so \(x \le -5\) or \(x \ge 3\). Both endpoints are included because \(\sqrt{0} = 0\) is defined.
The options with an open bracket at \(-5\) or 3 wrongly exclude a root, and \([-5, 3]\) is exactly where the expression is negative or zero.
Check with a value: at \(x = 0\) the expression is −15, so 0 must be excluded, and only the options without the interval around 0 survive.
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