Question

Which of the following is true for $y={\log }_{\sqrt{5}}\left(\sqrt{2}\right(\sin x-\cos x)+3)$

• A. $D:x\in RR:[0,2]$
• B. $D:x\in RR:[-2,0]$
• C. $D:x\in RR:[0,1]$
• D. $D:x\in RR:[-2,2]$

Answer 1

The correct option is A $D:x\in RR:[0,2]$

$y={\log }_{\sqrt{5}}\left(\sqrt{2}\right(\sin x-\cos x)+3)$
We know, $-\sqrt{2}\le \sin x-\cos x\le \sqrt{2}$
Hence domain is $x\in R$
and Range $\left[{\log }_{\sqrt{5}}\right(\sqrt{2}(-\sqrt{2}),{\log }_{\sqrt{5}}\left(\sqrt{2}\right(\sqrt{2})+3)]=[{\log }_{\sqrt{5}}1,{\log }_{\sqrt{5}}5]=[0,2]$