First slide

Functions (XII)

Question

The domain of the function $f (x) = \sqrt{e^{\sin^{- 1} (\log_{16} x^{2})}} is$

Moderate

A

$[\frac{1}{4}, 4]$
B

$[- 4, - \frac{1}{4}] \cup [\frac{1}{4}, 4]$
C

$[- 4, - \frac{1}{4}]$
D

None of these

Solution

f (x) is defined if
$\begin{array}{l} - 1 \leq \log_{16} x^{2} \leq 1 \Rightarrow 16^{- 1} \leq x^{2} \leq 16^{1} \\ \Rightarrow \frac{1}{16} \leq x^{2} \leq 16 \end{array} \begin{array}{l} x^{2} \geq \frac{1}{16} \Rightarrow (x - \frac{1}{4}) (x + \frac{1}{4}) \geq 0 \\ \Rightarrow x \leq - \frac{1}{4} or x \geq \frac{1}{4} . . . . . . . . (1) \end{array} x^{2} \leq 16 \Rightarrow (x - 4) (x + 4) \leq 0 \Rightarrow - 4 \leq x \leq 4 . . . . . . . (2)$
From Eq. (1) and (2), we get domain of f
$= [- 4, - \frac{1}{4}] \cup [\frac{1}{4}, 4]$

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