Question

如果-1＜x≤3求函数y=2x-1/x+1的取值范围

Answer 1

y = (2x-1)/(x+1) = (2x+2)/(x+1) + (-3)/(x+1) = 2 - 3/(x+1)
换句话说，本题已化减成为 1/(x+1) 的取值范围

-1 < x <= 3
0 < x+1 <= 4

取边界条件：
当 x+1 趋近於 0 时，1/(x+1) 趋近於无穷大
当 x+1 = 4 时，1/(x+1) = 1/4

故 y = 2 - 3/(x+1) 的取值范围为：
y <= 2 - 3/4 = 1.25

答：y <= 1.25

Answer 2

[注:可用反解法]
解:
∵y=(2x-1)/(x+1)
∴反解可得:x=(y+1)/(2-y)=[3/(2-y)]-1
∴x+1=3/(2-y).
∵-1＜x≦3
∴0＜x+1≦4.
即0＜3/(2-y)≦4
∴y≦5/4