若函数y=f(x)满足以下条件1、对于任意的x∈R,y∈R恒有f(x+y)=f(x)f(y);2、x∈（0,∞)时,f(x)∈（0,∞)(1)求f(0)的值；（2）求证f(x-y)=f(x)/(f(y)（f(y)≠0

Question

若函数y=f(x)满足以下条件1、对于任意的x∈R,y∈R恒有f(x+y)=f(x)f(y);2、x∈（0,∞)时,f(x)∈（0,∞)
(1)求f(0)的值；
（2）求证f(x-y)=f(x)/(f(y) （f(y)≠0

Answer 1

(1)
f(x+0)=f(x)f(0)=f(x)
因此f(0)=1
(2)
f(y-y)=f(y)f(-y)=f(0)=1
因此当f(y)≠0时,有
f(-y)=1/f(y)
因此
f(x-y)
=f(x)f(-y)
=f(x)/f(y)