Decompose $ f,g \colon A+B \to A'+B' $ into $ f \colon A \to A' $ and $ g \colon B \to B' $

Question

If I have a set x mapped to a set y 《1,5,2,3,4》>《0.84,-0.95,0.6,1,-0.75》 The two sets are actually the addition of

A =《1,5,4》>《0.84,-0.95,-0.75》 f (x)=sin (x)

And

B =《2,3》>《0.6,1.0》 f (x)=x/3

A and B are the input sets. If there was simply one unknown map there are a plethora of computational and numerical methods to find the mapping, but assuming we know there are two unknown maps, how can we determine which elements belong to which? I.e., decompose A+B to A and B,

any help appreciated. I am typing this on a android 4.4 phone with a very cracked screen, please excuse lack of latex.

Answer 1

I'm afraid you can't. Suppose the domain $\{0,1,2,3\}$ and corresponding values $\{0,1,0,1\}$.

There's no way to tell if it was originally
$f=0$ defined on $\{0,2\}$ and $g=1$ on $\{1,3\}$
or
$f(x)=x^3$ on $\{0,1\}$ and $g(x)=\log_{10}(9\,x-17)$ on $\{2,3\}$.