已知f(x)满足对任意x,y∈R,都有f(x+y)=f(x)乘f(y),且f(x)≠0,当x>0,f(x)>1,求证f(x)>0和f(x)在R上是增函数

Question

已知f(x)满足对任意x,y∈R,都有f(x+y)=f(x)乘f(y),且f(x)≠0,当x>0,f(x)>1,求证
f(x)>0和f(x)在R上是增函数

Answer 1

证明：f(0+0)=f(0)*f(0),f(0)=0或者f(0)=1.由f(x)≠0得,f(0)=1.
取x>0,f(x+(-x))=f(0)=f(x)*f(-x)=1,因为f(x)>1,所以0x2,那么f(x1-x2)>1,f(-x2)=1/f(x2),f(x1)*f(-x2)=f(x1)/f(x2)>1,所以f(x1)>f(x2).即f(x)在R上是增函数.