Discussion
Let \(f(x) = \sqrt{7 - x}\) and \(g(x) = \sqrt{x - 5}\). Then the domain of the function \(h(x) = f(x)g(x)\) is
Answer: A
The domain is [5,7].
h is defined where both square roots are defined. \(\sqrt{7 - x}\) needs \(x \le 7\), and \(\sqrt{x - 5}\) needs \(x \ge 5\).
\[ (-\infty, 7] \cap [5, \infty) = [5, 7] \]
\((-\infty, 7]\) and \([5, \infty)\) are the domains of f and g separately, not of their product. [-7,-5] has the signs reversed, and \((-\infty, \infty)\) ignores both conditions. Both end points are included because each root is 0 there.
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