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Find the domain of the function f(x) = sqrt( x*(x - 1)³*(x - 5) / ((x + 4)²*(x - 2))).

1. (-inf, -4) U (-4, 0] U [1, 2) U [5, inf)
2. [0, 1] U [5, inf)
3. (-4, 0] U [1, 2) U [5, inf)
4. [0, 1] U (2, 5]

Correct answer: (-inf, -4) U (-4, 0] U [1, 2) U [5, inf)

Solution

We need g(x) = x*(x-1)³*(x-5) / ((x+4)²*(x-2)) >= 0 with x != -4 and x != 2. Since (x+4)² > 0 (for x != -4) and (x-1)³ has the same sign as (x-1), the sign of g matches x*(x-1)*(x-5)/(x-2). Critical points: -4 (excluded, denominator), 0, 1, 2 (excluded), 5. Sign analysis of x*(x-1)*(x-5)/(x-2): for x < 0 (e.g. -1): (-)(-)(-)/(-) = (-)/(-) = +; between 0 and 1 (e.g. 0.5): (+)(-)(-)/(-) = (+)/(-) = -; between 1 and 2 (e.g. 1.5): (+)(+)(-)/(-) = (-)/(-) = +; between 2 and 5 (e.g. 3): (+)(+)(-)/(+) = -; x > 5 (e.g. 6): (+)(+)(+)/(+) = +. So g >= 0 on (-inf, 0] U [1, 2) U [5, inf), excluding x = -4. Include zeros 0, 1, 5; exclude 2 and -4. Domain = (-inf, -4) U (-4, 0] U [1, 2) U [5, inf).

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