The domain of the function
f(x) = sin ⁻¹( (3x² + x - 1)/(x - 1)² ) + cos ⁻¹( (x - 1)/(x + 1) ) is :

Solution

Maths sir · 4 steps

Each inverse function needs its argument between -1 and 1. Solve both conditions, then intersect them.

1. Condition for sin inverse. We need -1 ≤ (3x²+x-1)/(x-1)² ≤ 1 with x ≠ 1. Since (x-1)² > 0, multiply through without flipping any sign.
2. Solve both halves. Upper half: 3x²+x-1 ≤ x²-2x+1 gives (2x-1)(x+2) ≤ 0, so -2 ≤ x ≤ 1/2. Lower half: 3x²+x-1 ≥ -(x²-2x+1) gives x(4x-1) ≥ 0, so x ≤ 0 or x ≥ 1/4. Together this is [-2,0] ∪ [1/4,1/2].
3. Condition for cos inverse. We need -1 ≤ (x-1)/(x+1) ≤ 1. The right half becomes -2/(x+1) ≤ 0, so x > -1. The left half becomes 2x/(x+1) ≥ 0, so x ≥ 0 or x < -1. Both together give x ≥ 0.

   ([-2,0] ∪ [1/4,1/2]) ∩ [0,∞) = {0} ∪ [1/4,1/2]

4. So. The domain is [1/4,1/2] ∪ {0}. Check x = 0: both arguments equal -1, which is allowed. The answer is (c).

The idea to keep. For a sum of inverse trig functions, find each allowed set separately and take the intersection. Isolated points like {0} often survive, so check the endpoints.

Answer from the NTA final answer key. Solution by PrepWiser.