First slide

Functions (XII)

Question

Domain (D) and range (R) of $f (x) = \sin^{- 1} (\cos^{- 1} [x]),$ where [.] denotes the greatest integer function, is

Moderate

A

$D \equiv x \in [1, 2), R \equiv \{0\}$
B

$D \equiv x \in [0, 1], R \equiv \{- 1, 0, 1\}$
C

$D \equiv x \in [- 1, 1], R \equiv \{0, \sin^{- 1} (\frac{π}{2}), \sin^{- 1} (π)\}$
D

$D \equiv x \in [- 1, 1], R \equiv \{- \frac{π}{2}, 0, \frac{π}{2}\}$

Solution

$When [x] = 0, we have \sin^{- 1} (\cos^{- 1} 0) = \sin^{- 1} (π / 2), not defined .$
$When [x] = - 1, we have \sin^{- 1} (\cos^{- 1} (- 1)) = \sin^{- 1} (π), not defined .$
$When [x] = 1, we have \sin^{- 1} (\cos^{- 1} 1) = \sin^{- 1} (0) = 0 .$
$Hence, x \in [1, 2) and the range of function is \{0\} .$

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