Question

高一数学问题

设函数f(x)是定义在R上的增函数，且f(x)≠0,对于任意的x1,x2∈R，都有f(x1+x2)=f(x1)•f(x2).
(1)求证：f(x)>0
(2)求证：f(x1-x2)=f(x1)/f(x2) .

Answer 1

1.首先取x1=x2=x/2
f(x)=f(x/2)^2>=0
由于f(x)不=0,即f(x/2)不=0
所以有:f(x)>0
2.
f(x1)=f[x2+(x1-x2)]=f(x2)*f(x1-x2)
所以有:f(x1-x2)=f(x1)*(f(x2)

Answer 2

1.
f(x)=(f(x/2))^2
f(x)>0
2.
f(x+0)=f(x)f(0)
f(0)=1
f(x-x)=f(x)f(-x)
f(x)f(-x)=1
f(x1-x2)=f(x1)f(-x2)=f(x1)(1/f(x2))=f(x1)/f(x2)