bemath-1-Find-domain-of-function-f-x-log-0-2-x-2-x-1-1-2-sin-2-9pi-8-2x-sin-2-7pi-8-2x-sin-2018pi-4x-

Question Number 107304 by bemath last updated on 10/Aug/20

⋎ b e m a t h ⋎

(1) F i n d d o m a i n o f f u n c t i o n

f (x) = \sqrt{\log_{0.2} (\frac{x + 2}{x - 1}) - 1}

(2) \frac{\sin^{2} (\frac{9 π}{8} - 2 x) - \sin^{2} (\frac{7 π}{8} - 2 x)}{\sin (2018 π + 4 x)} = ?

Answered by bobhans last updated on 10/Aug/20

✠ bobhans ✠

(1) D_{f} : \log_{0.2} (\frac{x + 2}{x - 1}) - 1 ⩾ 0

\log_{0.2} (\frac{x + 2}{x - 1}) ⩾ \log_{0.2} (\frac{1}{5})

\frac{x + 2}{x - 1} ⩽ \frac{1}{5} \Rightarrow \frac{5 x + 10 - x + 1}{5 (x - 1)} ⩽ 0

\frac{4 x + 11}{5 (x - 1)} ⩽ 0 \Rightarrow - \frac{11}{4} ⩽ x < 1 \dots (1)

\Rightarrow \frac{x + 2}{x - 1} > 0 \Rightarrow x < - 2 \cup x > 1

solution \Rightarrow (1) \cap (2) : x ⩽ - \frac{11}{4}

Answered by bobhans last updated on 10/Aug/20

⌢ B obhans ⌢

(2) \frac{(\sin (\frac{9 π}{8} - 2 x) + \sin (\frac{7 π}{8} - 2 x)) (\sin (\frac{9 π}{8} - 2 x) - \sin (\frac{7 π}{8} - 2 x))}{\sin 4 x} =

\frac{(2 \sin (π - 2 x) \cos (\frac{π}{8})) (2 \cos (π - 2 x) \sin (\frac{π}{8}))}{\sin 4 x} =

\frac{\sin (\frac{π}{4}) (\sin (2 π - 4 x))}{\sin 4 x} = \frac{\frac{\sqrt{2}}{2} . \sin 4 x}{\sin 4 x} = \frac{\sqrt{2}}{2}

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