设函数f(x)的定义域为R,且满足下列两个条件:(1)存在x1≠x2,使f(x1)≠f(x2);(2)对任意x∈R,有f(x+y)=f(x)*f(y),(1)求f(0)
问题描述:
设函数f(x)的定义域为R,且满足下列两个条件:(1)存在x1≠x2,使f(x1)≠f(x2);(2)对任意x∈R,有f(x+y)=f(x)*f(y),(1)求f(0)
若令x=0,y≠0,则f(x+y)=f(y)=f(0)*f(y),所以f(0)=1.
怎么能保证f(y)不等于0
答
(1)令x=0,y≠0,则f(x+y)=f(y)=f(0)*f(y),所以f(0)=1.
(2)令x=y,则f(x+y)=f(2x)=f(x)^2>=0,
又因为存在x1≠x2,使f(x1)≠f(x2)且f(0)=1,
所以对任意x,y∈R,f(x)>0恒成立.