Question

函数f(x)对任意的实数x,y都满足f(xy)=f(x)+f(y),且f(1/3)=1,求f(9)

Answer 1

令y=1得

f(x) = f(x) + f(1)

∴f(1) = 0

令y = 1/x

f(1) = f(x) + f(1/x) = 0

f(x) = - f(1/x)

令 x=y

f(x²) = 2f(x)

∴f(9) = f(3²) = 2f(3) = 2* - f(1/3) = - 2