Question

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

• A. $[0,3]$
• B. $[-1,1]$
• C. $[0,1]$
• D. $[-1,3]$

Answer 1

The correct option is D $[-1,3]$

$f\left(x\right)=\sin x+\sqrt{3}\cos x+1=2(\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)+1$
$\Rightarrow f\left(x\right)=2(\cos {60}^0\sin x+\sin {60}^0\cos x)+1=2\sin (x+{60}^0)+1$
Now for $f\left(x\right)$ to be onto function. It's range and co-domain should be equal.
Thus S will be same as range of the given function which is $[-1,3]$
Since $-1\le \sin (x+{60}^0)\le 1$