Question

y=f(x)满足f(x)=f(2-x)恒成立,x>1时,f(x)为增函数,解不等式f(1-2x)>f(x)

Answer 1

f(x)=f(2-x)恒成立, y=f(x)的图像以x = (x + 2-x)/2 = 1为对称轴.

x>1时,f(x)为增函数, 则xf(x), 则 1-2x比x离直线x= 1更远, 即|1-2x -1| > | x - 1|

|2x| > | x - 1|

A. x ≥ 1:

2x > x -1

x > -1

结合前提,解为x ≥ 1

B. 0 ≤ x < 1:

2x > 1 - x

x > 1/3

结合前提,解为 1/3 < x < 1

C. x < 0:

-2x > 1 - x

-x > 1

x < -1

不等式f(1-2x)>f(x)的解为: x > 1/3 (来自A,B)或 x < -1 (来自C)