If $f: R → S$, defined by $f(x) = \sin x-\sqrt{3} \cos x + 1$, is onto, then the interval of S, is

[-1, 3]

We have,

$-\sqrt{1+(3)^2}≤\sin x-\sqrt{3} \cos x≤\sqrt{1+(3)^2}$ for all $x ∈ R$

$⇒-2≤\sin x-\sqrt{3} \cos x≤2$ for all $x ∈ R$

$⇒-1≤ \sin x-\sqrt{3} \cos x + 1 ≤3$ for all $x ∈ R$

$⇒-1≤f(x)≤\sqrt{3}$ for all $x ∈ R$

If $f: R → S$ is onto, then S = Range (f) = [-1, 3]