Finding multiple functions with same $f_{even}$ but different $f_{odd}$?

Question

A function can be decomposed as $f(x) = f_{even}(x) + f_{odd}(x)$ where $f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$.

If we know only $f_{even}$, how can we find different values for $f_{odd}$ that work (we can't just plug in any function)? A graphical method works too.

Answer 1

Let $f$ be your favorite function that isn't even. Then $f(x) \neq f_{even}(x)=(f_{even})_{even}(x)$, and so $f$ and $f_{even}$ are two different functions with the same even part.

Answer 2

There is a one to one correspondence between $A\times B=\{\text{even}\}\times \{\text{odd}\}$ and $C=\{\text{functions}\}$, given by $$ F(a,b) = a+b \\ G(f) = \left( x\to \frac{f(x) + f(-x)}2,x\to \frac{f(x) - f(-x)}2 \right) $$

In linear algebra terms, $C = A\bigoplus B$.

In other words, $f_{even}$ is not informative at all on $f_{odd}$.