Question

在计算机的算法语言中有一种函数[x]叫做取整函数（也称高斯函数）,它表示x的整数部分,即[x]是不超过x的最大整数．例如

Answer 1

f(x) = 2^x / (1 + 2^x) - 0.5

= 0.5 - 1/(1 + 2^x)

2^x > 0

0 < 1/(1 + 2^x) < 1

-0.5 < f(x) < 0.5

同理

-0.5 < f(-x) < 0.5

当 f(x) < 0

0.5 - 1/(1 + 2^x) < 0

=> x < 0

=> 0.5 - 1/(1 + 2^(-x)) > 0

f(-x) > 0

=> [f(x)] + [f(-x)] = -1 + 0 = -1

当 f(x) ≥ 0

0.5 - 1/(1 + 2^x) ≥ 0

=> x ≥ 0

=> 0.5 - 1/(1 + 2^(-x)) ≤ 0

f(-x) ≤ 0

=> [f(x)] + [f(-x)] = 0 - 1 = -1

所以 y = [f(x)] + [f(-x)] ∈ {-1}