Question

If $f:R\to S,definedbyf\left(x\right)=\sin x-\sqrt{3}\cos x+1$, is onto, then find the set $S$ ?

Answer 1

$f\left(x\right)=sinx\sqrt{3cosx}+1$

$2(\frac{1}{2}sinx-\frac{1}{2}\sqrt{3}cosx)+1$

$2\left(sin(x-\frac{\pi }{3})\right)+1$

$-1\le sinx-\frac{\pi }{3}\le 1$

$-2\le 2sinx-\frac{\pi }{3}\le 2$

$-1\le 2sinx-\frac{\pi }{3}+1\le 3$

$-1,3$