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Find the domain of the function f(x) = sqrt((2x + 1)/(x³ - 3x² + 2x)).

1. (-1/2, 0) U (1, 2)
2. [-1/2, 0) U (1, 2)
3. (-infinity, -1/2] U (0, 1) U (2, infinity)
4. (0, 1) U (2, infinity)

Correct answer: [-1/2, 0) U (1, 2)

Solution

We need (2x + 1)/[x(x - 1)(x - 2)] >= 0 with denominator != 0. Critical points: x = -1/2 (numerator zero), x = 0, 1, 2 (denominator zero, excluded). Sign analysis of g(x) = (2x+1)/[x(x-1)(x-2)]: for x < -1/2 the value is negative; on (-1/2, 0) positive; on (0, 1) negative; on (1, 2) positive; on (2, infinity) negative. Including x = -1/2 (where the whole expression = 0, allowed under sqrt) gives domain [-1/2, 0) U (1, 2).

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