Question

若f(x)和g(x)分别是奇函数与偶函数,且f(x)+g(x)=1/(x-1),则f(x),g(x)分别为多少

Answer 1

因为f(x)和g(x)分别是奇函数与偶函数
所以
f(-x) = -f(x)
g(-x) = g(x)
因为f(x) + g(x) = 1/(x - 1）
所以f(-x) + g(-x) = 1/(-x - 1)
-f(x) + g(x) = -1/(x + 1)
和原式相加得：
2g(x) = 2/(x² - 1)
g(x) = 1/(x² - 1)
所以 f(x) = x/(x² - 1)

Answer 2

f(x)+g(x)=1/(x-1) .....1
则f(-x)+g(-x) = 1/(-x-1)........2
因为f(x)和g(x)分别是奇函数与偶函数
所以 f(x)=-f(-x),g(x)=g(-x)
2式化为
-f(x) + g(x) = 1/(-x-1)........3
联立1 3，解得f(x) = x/(x^2-1)
g(x) = 1/(x^2-1)