Tell Mathematica that the output of a function is a List / Vector

Question

I have a function (Piecewise) that returns a list, e.g.:

F[x_] := Piecewise[{{{x^2, x}, x < 0}, {{x, x^2}, x > 0}}, {0, 0}];

I would like to add another list (e.g. {2,3}) to the output of that function, i.e., I would like to obtain:

If I simply do:

{2, 3} + F[x] 

I obtain:

{2 + F[x], 3 + F[x]} 

,i.e.:

which is not what I want. Naturally, I could include the list I want to add ({2,3}) in every branch of the definition of the Piecewise function, e.g.:

Piecewise[{{{2, 3} + {x^2, x}, x < 0}, {{2, 3} + {x, x^2}, x > 0}, {2,3}}]

But this is unsatisfactory, because my actual problem is much more convoluted that this MWE, with many different cases in the piecewise function, and I also want to use the result of the function as a vector in matrix multiplications and so on.

I have tried things like:

MapThread[Plus,{{2, 3}, F[x]}] 

Add[x_List, y_List] := x + y;
Add[{2, 3}, F[x]]

Assuming[VectorQ[F[_]], {2, 3} + F[x]]

without any success. Is there any simple way of doing this? Thanks a lot

Answer 1

As to your failed trial, I'm afraid you've severely misunderstood the usage of MapThread, _List and Assuming. Please notice Assuming only changes the global variable $Assumptions so it only has effect on those functions with Assumptions option; MapThread and _List simply don't do mathematical transforms, they won't handle Piecewise in the clever way you're imagining. (In my view, this post that you've also participated explains the topic well. )

As to your problem, the simplest solution I can think out is, transforming Piecewise to combination of UnitStep so the F[x] will be an explicit list. This can be easily done with Simplify`PWToUnitStep:

F[x_] = 
 Piecewise[{{{x^2, x}, x < 0}, {{x, x^2}, x > 0}}, {0, 0}] // Simplify`PWToUnitStep
(* {x (1 - UnitStep[-x]) + x^2 (1 - UnitStep[x]), 
 x^2 (1 - UnitStep[-x]) + x (1 - UnitStep[x])} *)

trial = F[x] + {2, 3};

Plot[trial, {x, -4, 4}]