Question

设fx是定义在负无穷到零并上零到正无穷上的函数满足fx y=fx+fy当x大于一时fx

设f(x)是定义在0到正无穷大上的增函数且对一切x.y>0满足f(x/y)=fx-fy若f（2）=1,解不等式f(x+3)-f(1/x)<2

Answer 1

(1)
∵对一切x.y>0满足f(x/y)=fx-fy
∴当x=y时f(x/y）=f(x/x）=f(1)=fx-fx=0
∴f(1)=0
(2)
∵f(x/y)=fx-fy
f(x+3)-f(1/x)<2
f(x²+3x）＜2f(2)
f(x²+3x）＜f(4)
x²+3x＜4
x²+3x-4＜0
(x+4)(x-1)＜0
-4＜x＜1