Provided 0 is a part of the domain of f(x)
f(0)=-f(-0)=>
f(0)=-f(0), (-0=0 obviously)
2f(0)=0
f(0)=0
This immediately implies if f(x) is an odd polynomial, 0 is a root, (x-0)=x is a factor (factor theorem), and hence f(x) is divisible by x.
To answer the f(x)=1/x question, it is a...