Let f(x) be any function with domain being all real x.
The functions E(x) and O(x) are defined as: $$E(x)=(f(x)+f(-x))/2$$ $$O(x)=(f(x)-f(-x))/2$$ Also, $$f(x)=E(x)+O(x)$$ Investigate the importance of the domain (all real x) for f(x) and give a few examples demonstrating what can go wrong if the domain is not all real x.
The domain does not have to be all real $x$. For example, your domain could just be $\{0\}$ and you could happily have your functions as $f(0) = 0$, $E(0) = 0$, $O(0) = 0$ (or maybe something else).
However, when you have a system of equations like this, it is more interesting to ask what your domain could be. In particular, on how large of a domain can you have functions which satisfy these equalities? Can you find functions which satisfy these equalities for all real $x$? Maybe if you require the domain to be all real $x$ you get a unique solution? These are questions that make the concept of the domain useful.
In this case, with a proper domain, these equations will hold true regardless of the $f$ you choose (by defining $E$ and $O$ as you did. However, a proper domain is crucial. For example, if you choose the domain of $f$ to be $[0,1]$, then you can't define $E(1)$, since $f(-1)$ is not defined! You will need your domain to be symmetric around $0$ (though not necessarily all real $x$), such as $[-1,1]$.