Question

Find the domain of definition and the range of the following function:
$f\left(x\right)={\log }_2\frac{\sin x-\cos x+3\sqrt{2}}{\sqrt{2}}.$

Answer 1

$f\left(x\right)={\log }_2\left(\sin (x-\frac{\pi }{4})+3\right)$

$\sin (x-\frac{\pi }{3})+3>0-1<\sin (x-\frac{\pi }{3})<12<\sin (x-\frac{\pi }{3})+3<4x\in R$

Domain $x\in R$

and

$k=\sin (x-\frac{\pi }{3})+3$

Minimum value of $k=2$

Maximum value of $k=4$

$f\left(x\right)\in [1,2]$

Range $\in [1,2]$