First slide

Properties of ITF

Question

The range of the function $f (x) = \sin^{- 1} [x^{2} + \frac{1}{2}] + \cos^{- 1} [x^{2} - \frac{1}{2}]$ where $[]$ denotes greatest integer function

Moderate

A

${\frac{π}{2}}$
B

${π}$
C

${\frac{- 1}{2}, 0}$
D

${0, \frac{π}{2}}$

Solution

$[x^{2} - \frac{1}{2}] = [x^{2} + \frac{1}{2} - 1] = [x^{2} + \frac{1}{2}] - 1$
$f (x) = \sin^{- 1} [x^{2} + \frac{1}{2}] + \cos^{- 1} ((x^{2} + \frac{1}{2}) - 1)$

$[x^{2} + \frac{1}{2}] = 0, 1$

$f (x) = \sin^{- 1} (0) + \cos^{- 1} (- 1)$ or $\sin^{- 1} (1) + \cos^{- 1} (0)$

$∴ f (x) = π$

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