Question

The range of $f\left(x\right)={\log }_{\sqrt{5}}\{\sqrt{2}(\sin x-\cos x)+3\}$ is

• A. $[0,\sqrt{2}]$
• B. $[0,2]$
• C. $[\sqrt{2},2]$
• D. $[-\sqrt{2},\sqrt{2}]$

Answer 1

The correct option is B $[0,2]$

Let, $y={\log }_{\sqrt{5}}\{\sqrt{2}(\sin x-\cos x)+3\}$

or, $\sqrt{2}(\sin x-\cos x)=(\sqrt{5}{)}^y-3$

or, $2\sin (x-\frac{\pi }{4})=(\sqrt{5}{)}^y-3$

We have $-2\le |2\sin (x-\frac{\pi }{4})|\le 2$

or, $-2\le (\sqrt{5}{)}^y-3\le 2$

or, $1\le (\sqrt{5}{)}^y\le 5$

or, $0\le y\le 2$

$\therefore$ range of $f\left(x\right)$ is $[0,2]$.