Function Inequality?

Question

I have f(x)=f(2x). How do I find the polynomials (or lack of) that make this true (or false). The important thing is how do I prove it?

Answer 1

Use the general equation of a polynomial
f(x) = a0 + a1 x + a2 x^2 + ... + an x^n where a0 through an are constants.
Now find f(2x)
f(2x) = a0 + a1 2x + a2 2^2 x^2 + ... + an 2^n x^n

Compare the terms in each equation. For positive x, each term of f(2x) is greater than the corresponding term of f(x) (other than the contant term which is equal). For negative x, each term is less. So if any of the constants other than a0 is non-zero, f(2x) will be greater than or less than f(x) for all x not equal to zero. So all of the constants but a0 must be zero for them to equal for all x.

Let f(x) = c for some constant c.
then f(x) = f(2x)

Answer 2

it's 0