First slide

Functions (XII)

Question

The domain of the function $f (x) = \log_{e} \{\log_{|\sin x|} (x^{2} - 8 x + 23) - \frac{3}{\log_{2} |\sin x|}\}$ contains which of the following interval(s)?contains which of the following interval(s)?

Moderate

A

$(3, π)$
B

$(π, \frac{3 π}{2})$
C

$(\frac{3 π}{2}, 5)$
D

None of these

Solution

$\begin{array}{l} f (x) is defined if \log_{|\sin x|} (x^{2} - 8 x + 23) - \frac{3}{\log_{2} |\sin x|} > 0 \\ or \log_{|\sin x|} (\frac{x^{2} - 8 x + 23}{8}) > 0 \\ This is true if |\sin x| \neq 0, and \frac{x^{2} - 8 x + 23}{8} < 1 . \\ Now, \frac{x^{2} - 8 x + 23}{8} < 1 or x^{2} - 8 x + 15 < 0 \\ or x \in (3, 5) - \{π, \frac{3 π}{2}\} \\ Domain = (3, π) \cup (π, \frac{3 π}{2}) \cup (\frac{3 π}{2}, 5) \end{array}$

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