Question

定义域为R的两个函数f(x)和g(x),对于任意x,y∈R满足：f(x-y)=f(x)g(y)-g(x)f(y)且f（1

Answer 1

令x=y,①式即

f(0)=f(x)g(x)-g(x)f(x),

所以 f(0)=0 ③

令x=1,y=1, ①式即

f(1)=f(1)g(0)-g(1)f(0),

由③ f(1)=f(1)g(0),

再由② g(0)=1 ④,

令x=0,①式即

f(-y)=f(0)g(y)-g(0)f(y),

由③④ f(-y)=-f(y),所以f(x)是奇函数.

3.