Question

Determine range of the function $y={\log }_2\left[\frac{\sin x-\cos x+3\sqrt{2}}{\sqrt{2}}\right]$

• A. $[-1,2]$
• B. $[0,2]$
• C. $[-1,0]$
• D. $[1,2]$

Answer 1

The correct option is D $[1,2]$

$y={\log }_2\left\{\frac{\sqrt{2}\sin (x-\frac{\pi }{4})+3\sqrt{2}}{\sqrt{2}}\right\}$ or $y={\log }_2\{\sin (x-\frac{\pi }{4})+3\}$ ...(1)Now $-1\le \sin (x-\frac{\pi }{4})<1\forall xϵR$
$\therefore 3-1\le 3+\sin (x-\frac{\pi }{4})<4\forall xϵR$ Hence domain is R.For range we have $2^y=\sin (x-\frac{\pi }{4})+3$ by(1) or $2^y-3=\sin (x-\frac{\pi }{4})$ or $-1\le 2^y-3\le 1$ as $-1\le \sin \theta \le 1$ or $2\le 2^y\le 4$ or $2^1\le 2^y\le 2^2$
$\therefore 1\le y\le 2$ Hence range is [1, 2].