Specify a domain of inversion for an inverse function

Question

I'm playing around with some functions related to the Lambert W Function. Namely those of the form:

$$W(x;a) = f^{-1}\left(x \left( e^{x} - a\right) \right)$$

And when $a$ gets too large, Mathematica begins to detect the lower branch as the principle.

plogext[x_, a_] := x*(Exp[x] - a)
Animate[Plot[{ProductLog[x], 
  InverseFunction[plogext @@ {#1, a} &][x]}, {x, -1, 10}], {a, 0, 3, 0.001}]

I can't figure out how to tell Mathematica that the top branch is more interesting. I did try changing the definition of plogext to have a condition. But in that case, it simply fails. (This bound only holds for $a > -e^{-2}$).

plogext[x_, a_] := x*(Exp[x] - a) /; x >= (-1 + ProductLog[a E])
Animate[Plot[{ProductLog[x], 
  InverseFunction[plogext @@ {#1, a} &][x]}, {x, -1, 10}], {a, 0, 3, 0.001}]

Answer 1

plogext01= ConditionalExpression[ #1 (Exp[#1] - #2), #1>0]&
Manipulate[Plot[{
x,
plogext01[x, a], 
InverseFunction[plogext01 @@ {#1, a} &][x]
  }, {x, -2, 2},PlotRange-> 2{{-1,1},{-1,1}},PlotStyle-> {Red,Green,Blue}], 
  {a, 0, 3, 0.001,Appearance-> "Open"}]  

Compare with your code (slightly modifed) :

plogext= #1 (Exp[#1] - #2) &
list02=Manipulate[Plot[{
x,
plogext[x, a], 
InverseFunction[plogext @@ {#1, a} &][x]
  }, {x, -2, 2},PlotRange-> 2{{-1,1},{-1,1}},PlotStyle-> {Red,Green,Blue}], 
  {a, 0, 3, 0.01,Appearance-> "Open"}]