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Find the domain of: (i) f(x) = log_(2x)(x² - 5x + 13); (ii) f(x) = sqrt((9 - 3^x)/(5^(-x) - 125)).

1. (i) x>0, x≠1/2; (ii) x ∈ (-3, 2]
2. (i) x>0; (ii) x ∈ [2, ∞)
3. (i) x≥0; (ii) x ∈ (-3, 2)
4. (i) all real x; (ii) x ∈ (2, 3)

Correct answer: (i) x>0, x≠1/2; (ii) x ∈ (-3, 2]

Solution

(i) Argument x² - 5x + 13 has discriminant 25 - 52 < 0, so it is always positive: no constraint from it. Base 2x needs 2x>0 (x>0) and 2x≠1 (x≠1/2). Domain: x>0, x≠1/2. (ii) Need (9 - 3^x)/(5^-x - 125) >= 0. Numerator 9 - 3^x >= 0 ⇒ 3^x <= 9 ⇒ x <= 2. Denominator 5^-x - 125 = 5^-x - 5³; 5^-x - 125 > 0 ⇒ -x > 3 ⇒ x < -3. Sign analysis of the quotient gives the radicand >= 0 on (-3, 2], excluding x=-3 (denominator zero) and including x=2 (numerator zero). Domain: (-3, 2].

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