The range of the function y=2sin- 1 [x2 + (1/2)]+cos- 1â¡[x2 - (1/2)] is (where, [⋅ ] denotes the greatest integer function)

Question

The range of the function y=2sin- 1 [x2 + (1/2)]+cos- 1â¡[x2 - (1/2)] is (where, [⋅ ] denotes the greatest integer function) (A) (0 , π ) (B) [π , (3 π /2)] (C)  π  (D)  π , (3 π /2)

Tardigrade Admin · Accepted Answer

Let, [x2 - (1/2)]=[x2 + (1/2) - 1]=α ∈ I ⇒ [x2 + (1/2)]-1=α ∴ y=2(sin)- 1 (α + 1)+(cos)- 1â¡α , α = - 1,0 At α =-1 y=2(sin)- 1 (0)+(cos)- 1 â¡ (- 1)=π At α =0 y=2(sin)- 1 (1)+(cos)- 1 â¡ (0)=(3 π /2)

Q. The range of the function $y = 2 s i n^{- 1} [x^{2} + \frac{1}{2}] + co s^{- 1} [x^{2} - \frac{1}{2}]$ is (where, $[\cdot]$ denotes the greatest integer function)

$(0, π)$

$[π, \frac{3 π}{2}]$

${π}$

${π, \frac{3 π}{2}}$

Q. The range of the function y=2sin−1[x2+21​]+cos−1⁡[x2−21​] is (where, [⋅] denotes the greatest integer function)

(0,π)

[π,23π​]

{π}

{π,23π​}

Let, [x2−21​]=[x2+21​−1]=α∈I ⇒[x2+21​]−1=α ∴y=2(sin)−1(α+1)+(cos)−1⁡α,α={−1,0} At α=−1 y=2(sin)−1(0)+(cos)−1⁡(−1)=π At α=0 y=2(sin)−1(1)+(cos)−1⁡(0)=23π​

Q. The range of the function $y = 2 s i n^{- 1} [x^{2} + \frac{1}{2}] + co s^{- 1} [x^{2} - \frac{1}{2}]$ is (where, $[\cdot]$ denotes the greatest integer function)

$(0, π)$

$[π, \frac{3 π}{2}]$

${π}$

${π, \frac{3 π}{2}}$