求函数.y=x+4根号（1-x) 和y=x+根号(1-x^2)的值域

1)设√(1-x)=t(0<=t<=1),则x=1-t²

y=1-t²+4t=-(t-2)²+5是关于t的二次函数,对称轴t=2,开口向下

∴最小值在t=-1处取得,为-3²+5=-4;最大值在t=1处取得,为-1²+5=4

∴值域为[-4,4]

2)设x=cost(t∈[0,π]),∴sint>0,√(1-x²)=√(1-cos²t)=√(sin²t)=sint

∴y=sinx+cosx=√2sin(x+π/4)

∵-1<=sin(x+π/4)<=1

∴-√2<=√2sin(x+π/4)<=√2,即-√2<=y<=√2

值域为[-√2,√2]