If the domain of the function $f(x) = \log(1 - \log(x^2 - 9x + 18))$ is $(\alpha, \beta) \cup (\gamma, \delta)$, then $\alpha + \beta + \gamma + \delta$ is equal to
Step-by-Step Solution
Key Concept: Impose every domain restriction from logarithms, roots, denominators, and inverse functions before intersecting intervals.
Domain: $1 - \log(x^2 - 9x + 18) > 0$ ... (1)
Also, $x^2 - 9x + 18 > 0$
$(x - 3)(x - 6) > 0$
$x \in (-\infty, 3) \cup (6, \infty)$ ... (i)
Also, $x^2 - 9x + 18 < 4$
$x^2 - 9x + 14 < 0$
$(x - 2)(x - 7) < 0$
$x \in (2, 7)$ ... (ii)
(i) $\cap$ (ii): $(2, 3) \cup (6, 7) = (\alpha, \beta) \cup (\gamma, \delta)$
$\Rightarrow \alpha + \beta + \gamma + \delta = 2 + 3 + 6 + 7 = 18$
<div class="key-concept"><strong>Key Concept:</strong> Impose every domain restriction from logarithms, roots, denominators, and inverse functions before intersecting intervals.</div>
<div class="trap-box"><strong>Trap:</strong> Do not solve only the outermost condition; every nested function contributes its own restriction.</div>
Correct Answer: 1