If f: ℝ arrow S defined by f(x)=4 x-3 x+1 is onto, then what is S...

$f(x)=4 \sin x-3 \cos x+1$ To find range of this function, we have to find maximum and minimum values. We know, for a $\sin \theta+\mathrm{b} \cos \theta+\mathrm{c}$, max value is $\sqrt{\mathrm{a}^2+\mathrm{b}^2}+\mathrm{c}$ and minimum value is $-\sqrt{\mathrm{a}^2+\mathrm{b}^2}+\mathrm{c}$. $$\begin{aligned} \therefore \text{ Maximum value } & =\sqrt{(4)^2+(-3)^2}+1 \\ & =\sqrt{16+9}+1=5+1=6 \end{aligned}$$ Minimum value $=-\sqrt{16+9+1}=-5+1=-4\therefore$ Range, $\mathrm{S} \in[-4,6]$