Question

(本小题满分10分)
已知函数，当点(x，y) 是函数y = f (x) 图象上的点时，点是函数y = g(x) 图象上的点．
(1)   写出函数y = g (x) 的表达式；
(2)   当g(x)-f (x)0时，求x的取值范围；
(3)   当x在(2) 所给范围内取值时，求的最大值．

Answer 1

（1）
（2）
（3）[g(x)-f (x)]_max=

解得0 ≤ x ≤ 1；·········································································· 6分
(3) 当0≤ x≤ 1时，g(x) -f (x) =-log ₂(x+1)，
令，
则kx² + (2k-3)x + (k-1) = 0，························································· 7分
∵x的取值存在，∴D = (2k-3)²-4k(k-1) ≥ 0，
解得：，·········································································· 9分
当k=时，x=∈[0，1]，
∴当x=时，[g(x)-f (x)]_max=．········································· 10分