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Find the domain of the function f(x) = sin⁻¹((3x² + x - 1) / (x - 1)²) + cos⁻¹((x - 1) / (x + 1)).

1. [0, 1/4]
2. [-2, 0] union [1/4, 1/2]
3. [1/4, 1/2] union {0}
4. [0, 1/2]

Correct answer: [1/4, 1/2] union {0}

Solution

The cos⁻¹ term requires (x-1)/(x+1) in [-1,1], which forces x >= 0 (with x != -1). The sin⁻¹ term requires (3x²+x-1)/(x-1)² in [-1,1], which gives x in [-2,1/2] intersected with (x <= 0 or x >= 1/4), excluding x = 1. Intersecting with x >= 0 yields {0} union [1/4, 1/2].

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