求函数f(x)=2(sinx+cosx)+sin2x-1的值域

f(x)=2(sinx+cosx)+sin2x-1 =2√(sin2x+1)+(sin2x+1)-2 =[√(sin2x+1)+1]²-3∵0≤sin2x+1≤2∴0≤√(sin2x+1)≤√2∴1≤√(sin2x+1)+1≤√2+1∴1≤[√(sin2x+1)+1]²≤3+2√2-2≤[√(sin2x+1)+1]²-3≤2√2∴f(x)∈[-2,2√2]