Question

The range of the function f(x) = sin^-1[x² + 1/2] + cos^-1[x² - 1/2] where [·] is the greatest integer function, is

(a)  {π/2,π}

(b)  {0,π/2,}

(c)  {π}

(d)  {0,π/2}

Answer 1

Correct option  (c)  {π}

Explanation :

We have

 Since x² + 1/2 ≥ 1/2, we get

x² + 1/2  ≥ 1/2, we get

[x² + 1/2] = 0 or 1

Since sin⁻¹ [x² + (1/2)] is defined only for these two values,

(i)   when [ (x² + (1/2)]  = 0, we get

f(x) = sin⁻¹0 + cos⁻¹(−1) = (ii) when [ ( x / )] 2 + 1 2 = 1, we get f(x) = sin−11 + cos−10 = 

(ii) when [x² + (1/2)] = 1, we get 

f(x) = sin⁻¹1 + cos⁻¹0 = π

Therefore, the range of f(x) = {π}.