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Question: Range of f(x) = sin<sup>−1</sup>\(\left\lbrack x^{2} + \frac{1}{2} \right\rbrack + \cos^{- 1}\left\l...

Range of f(x) = sin−1[x2+12]+cos⁡−1[x2−12]\left\lbrack x^{2} + \frac{1}{2} \right\rbrack + \cos^{- 1}\left\lbrack x^{2} - \frac{1}{2} \right\rbrack,where [.] denotes the greatest integer function, is

A

{π2,π}\left\{ \frac{\pi}{2},\pi \right\}

B

{π}

C

{π2}\left\{ \frac{\pi}{2} \right\}

D

None of these

Answer

{π}

Explanation

Solution

[x2+12]=[x2−12+1]=1+[x2−12]\left[ x ^ { 2 } + \frac { 1 } { 2 } \right] = \left[ x ^ { 2 } - \frac { 1 } { 2 } + 1 \right] = 1 + \left[ x ^ { 2 } - \frac { 1 } { 2 } \right] .

Thus for domain point of view.

[x2−12]=0,1⇒[x2+12]\left[ x ^ { 2 } - \frac { 1 } { 2 } \right] = 0,1 \Rightarrow \left[ x ^ { 2 } + \frac { 1 } { 2 } \right] = 1, 0

⇒ f(x) = sin-1(1) + cos-1(0)

or sin-1(0) + cos-1(-1)

⇒ f(x) = {π}