$Therangeofthefunctionf\left(x\right)=[\sin x+\cos x]\left(where\left[x\right]denotesthegreatest\mathrm{int}egerfunction\right)is$

$\begin{array}{l}Therangeof\sin x+\cos x=[c-\sqrt{a^2+b^2},c+\sqrt{a^2+b^2}]\\ =[0-\sqrt{1+1},0+\sqrt{1+1}]\\ =[-\sqrt{2},\sqrt{2}]\\ \therefore -\sqrt{2}\le \sin x+\cos x\le \sqrt{2}\\ \therefore Rangeoff\left(x\right)=[\sin x+\cos x]=\{-2,-1,0,1\}\end{array}$