fa.functional analysis – Decomposition of a function into right-sided and left-sided function

Here I define a distribution $fin D’$ to be right-sided if supp $fsubseteq (0,infty)$ and defnote it by $f_+$ and if the supp $fsubseteq (-infty,0)$ it is called left-sided and denoted by $f_-$.

Now, it is claimed that if $f$ is locally integrable function on $mathbb{R}$, then there is a unique decomposition $f=f_++f_-$ where $f_+$ is right-sided locally integrable function and $f_-$ is left sided locally integrable function.

For an example:

If I have $A(omega)=frac{1}{omega^2+9}$
then I can find a decomposition $A_+(omega)=frac{i}{6(omega+3i)}$ and $A_-(omega)=frac{-i}{6(omega-3i)}$ by inspection.

But, How do I find such a decomposition for function like
$e^{-a x}theta(-x)$ where $theta$ is Heaviside step function? Is there a general process to find such a decomposition?