Solving a transcendental function using the Lambert function The solution to the equation
$$Xe^X=K$$
is given by 
$$X=W(K)$$
where $W$ is the Lambert function.
This idea was extended here to show that the solution to
$$\frac{1-e^X}{X}=K?$$
is given by 
$$X = \frac 1K-W\left(\frac{e^{1/K}}K\right). $$
My question is, can we use a similar method to determine the solution to
$$\frac{\left(1 - e^{-X}  \right)}{X} - e^X = K?$$
 A: For solving a given equation $F(x)=c$, $c$ constant, by isolating $x$ on one side of the equation, you can transform the equation if you can apply a suitable partial inverse $F^{-1}$ of the function $F$: $x=F^{-1}(c)$. In your equation
$$\frac{1-e^{-x}}{x}-e^{x}=k,$$
the left-hand side is an elementary function. It is an algebraic function that depends on the algebraic independent functions $x$, $e^{-x}$, $e^{x}$. It cannot be transformed into an algebraic equation of only one algebraic independent elementary function. According to a theorem of Ritt, such an elementary function cannot have an inverse that is also an elementary function. Therefore you cannot isolate $x$ by applying only elementary operations (elementary functions) to this equation.
We consider now the elementary functions and Lambert W function. To apply Lambert W, its inverse $xe^{x}$ is necessary. Together with the elementary functions, an equation
$$f_{1}(f_2(x)e^{f_2(x)})=c\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
with constant $c$ is needed where $f_1$ and $f_2$ are elementary functions with a suitable elementary partial inverse.
We therefore transform:
$$1-e^{-x}-xe^{x}=kx,$$
$$1-kx-e^{-x}-xe^{x}=0.$$
The left-hand side of the last equation is an algebraic function that depends on the algebraic independent functions $x$, $e^{-x}$, $xe^{x}$. Observing the form of equation (1), the elementary functions $f_1$ and $f_2$ cannot be transformed into algebraic functions of only one algebraic independent elementary function. Therefore you cannot isolate $xe^{x}$ by applying only elementary operations (elementary functions). Therefore you cannot apply Lambert W and only elementary operations (elementary functions) to isolate $x$ from your equation.
