Question

若非零函数f(x)对任意实数a,b均有f(a+b)=f(a)f(b),且当x小于零时,f(x)大于

1 (1)求证f(x)>0;(2)若f(4)=16,求f(1)的值；(3)求证f(x)在其定义域内为减函数；(4)当f(4)=16时，解不等式f(x-3)f(7)小于或等于4

Answer 1

(1)令a=b=x/2f(x)=f(x/2)*f(x/2)=[f(x/2)]^2非零函数f(x)所以f(x)>0(2)令a=x1-x2 b=x2 且x1<x2f(x1-x2+x2)=f(x1-x2)*f(x2)f(x1)=f(x1-x2)*f(x2)[x1-x2<0 f(x1-x2)>1且f(x1)>0 f(x2)>0]f(x1)/f(x2)>1f(x1)>f(x2)即得当x1<x2 f(x1)>f(x2)所以f(x)为减函数(3)f(4)=f(2)*f(2) f(2)>0所以f(2)=1/4f(x-3)*f(5-x^2)<=1/4f(x-3+5-x^2)<=f(2)[f(x)为减函数]x-3+5-x^2>=2x^2-x<=00<=x<=1