Relationship between two random variables having same expectation

[omaradib]

Homework Statement

Say, it is known that
$$E_X[f(X)] = E_X[g(X)] = a$$ where $$f(X)$$ and $$g(X)$$ are two functions of the same random variable $$X$$. What is the relationship between $$f(X)$$ and $$g(X)$$?

Homework Equations

The Attempt at a Solution

My answer is $$f(X) = g(X) + h(X)$$ where $$E_X[h(X)] = 0$$.

This relationship is apparent by taking expectation in both sides. But, is it necessarily and sufficiently true?

 

[omaradib]

Anyone please?