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

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 $f_{odd}$.
