Question

已知x,y属于R,且x的绝对值小于1,(x-y)/(xy-1)的绝对值小于等于1,则y的取值范围是

Answer 1

|(x-y)/(xy-1)|≤1

即|x-y|≤|xy-1|且xy≠1

即|x-y|²≤|xy-1|²,且xy≠1

y²-2xy+x²≤x²y²-2xy+1,且xy≠1

(1-x²)(y²-1）≤0,且xy≠1

因为|x|＜1,于是1-x²＞0

于是y²≤1,此时y∈[-1,1]

|xy|＜|y|=1,满足xy≠1

于是y得范围是[-1,1].