Obtain a function by it's even or odd part

Question

Since every function can be divided in a even and a odd part:

\begin{equation} f_e(x)=\frac{f(x)+f(-x)}{2} \end{equation}

\begin{equation} f_o(x)=\frac{f(x)-f(-x)}{2} \end{equation}

\begin{equation} f(x)=f_e(x)+f_o(x) \end{equation}

How can I obtain the function by it's even part? Or it's odd part?

I mean, having $f_e(x)$ obtain $f(x)$ ou having $f_o(x)$ obtain $f(x)$

Answer 1

You can't obtain $f$ from just $f_\mathrm{e}$ or just $f_\mathrm{o}$. Information gets lost. For example take a function $f$ and then define $g(x)=f(x)+\sin(x)$ and $h(x)=f(x)+\cos(x)$, then we have $g_\mathrm{e}(x)=f_\mathrm{e}(x)$ and $g_\mathrm{o}(x)=f_\mathrm{o}(x)+\sin(x)$ as well as $h_\mathrm{e}(x)=f_\mathrm{e}(x)+\cos(x)$ and $h_\mathrm{o}(x)=f_\mathrm{o}(x)$. As you can see the even parts can be identical while the odd parts aren't and vice versa.

Answer 2

You can't. Just consider for example even functions : they have the function $0$ as odd part so you can't recover the function from the function $0$.

Answer 3

Imagine you could express $f(x)=\phi(f_e(x))$

Then it is clear that $f(-x)=\phi(f_e(-x))=\phi(f_e(x))=f(x)\implies f=f_e$

Same for the odd part.

This means that unless $f$ is even or odd to start with, you cannot express $f$ only in function of $f_e$ or $f_o$, you need both.