First slide

Functions (XII)

Question

$f (x) = \sin^{- 1} [e^{x}] + \sin^{- 1} [e^{- x}], where [.] greatest integer function then$

Difficult

A

$domain of f (x) is (- \log_{e} 2, \log_{e} 2)$
B

$range of f (x) = \{π\}$
C

$Range of f (x) is \{\frac{π}{2}, π\}$
D

$f (x) = \cos^{- 1} x has only one solution$

Solution

$\begin{array}{l} We have f (x) = \sin^{- 1} [e^{x}] + \sin^{- 1} [e^{- x}] \\ we must have 0 < e^{x} < 2 and 0 < e^{- x} < 2 \\ \Rightarrow x \in (- \log_{e} 2, \log_{e} 2) \\ \Rightarrow f (x) = \{\begin{cases} π, x = 0 \\ \frac{π}{2}, x \in (- \log_{e,} 2, 0) \cup (0, \log_{e} 2) \end{cases} \end{array}$

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