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Find the domain of each function: (i) y = 1/sqrt(log_(x+1)(x - 1)) (ii) y = sqrt(logₓ(x² - 1)) (iii) y = log_(x-2)((x² - 4)/(x + 1))

1. (i) x > 2; (ii) x > sqrt(2); (iii) x in (2,3) U (3, infinity)
2. (i) x > 1; (ii) x > 1; (iii) x > 2
3. (i) x > 2; (ii) x > 1; (iii) x in (2, infinity)
4. (i) no solution; (ii) x > sqrt(2); (iii) x in (3, infinity)

Correct answer: (i) x > 2; (ii) x > sqrt(2); (iii) x in (2,3) U (3, infinity)

Solution

Each part combines base restrictions, argument-positivity, and the sign of the logarithm. Solving the simultaneous conditions yields the listed domains.

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