First slide

Functions (XII)

Question

$Range of the function f (x) = \sqrt{\cos^{- 1} (\sqrt{\log_{4} x}) - \frac{π}{2}}$ $+ \sin^{- 1} (\frac{1 + x^{2}}{4 x}) is$

Difficult

A

$(0, \frac{π}{2} + \sqrt{\frac{π}{2}}]$
B

$[\frac{π}{2}, \frac{π}{2} + \sqrt{\frac{π}{2}}]$
C

$[\frac{π}{6}, \frac{π}{2}]$
D

$\{\frac{π}{6}\}$

Solution

$f (x) = \sqrt{- \sin^{- 1} (\sqrt{\log {}_{4}x})} + \sin^{- 1} (\frac{1 + x^{2}}{4 x})$ $(\sin c e \sin^{- 1} x + \cos^{- 1} x = \frac{π}{2})$
$d o m a i n o f f i r s t f u n c t i o n i s \{1\} o n l y$ $\{\sin c e - \sin^{- 1} (\sqrt{\log {}_{4}x}) \geq 0 \Rightarrow \sin^{- 1} (\sqrt{\log {}_{4}x}) \leq 0 \Rightarrow \sqrt{\log {}_{4}x} \leq \sin 0 \Rightarrow \log {}_{4}x = 0 \Rightarrow x = 1\}$
$t h e r e f o r e f (x) d o m a i n a l s o \{1\}$ $(\sin c e D o m a i n o f (f + g) = (D o m a i n o f f) \cap (D o m a i n o f g))$
$R a n g e o f f (x) = \{f (1)\} = \{\sqrt{- \sin^{- 1} 0} + \sin^{- 1} (\frac{1 + 1}{4})\} = \{\sin^{- 1} \frac{1}{2}\} = \{\frac{π}{6}\}$

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