Question

高一函数题:若非零函数f(x)对任意实数a,b均有f(a+b)=f(a)乘f(b),且当x<0时,f(x)>1； 求证：f(x)>0

高一函数题:若非零函数f(x)对任意实数a,b均有f(a+b)=f(a)乘f(b),且当x<0时,f(x)>1；
求证：f(x)>0

Answer 1

f(a+b) = f(a)f(b)
put a=b=0
f(0) = f(0)f(0)
=> f(0) = 1

case 1: for x <0
f(x) >1 >0 (true)
case 2: x=0
f(0) =1 >0 (true)
case 3: for x>0
-x <0
f(0) = f(x+(-x)) = f(x)f(-x)
=> f(x) > 0 ( f(-x) >0 , f(0) =1 )