Generalization of Lambert W function?

Question

Can the function $f(x)$ defined by $$ x = f(x)^2 e^{f(x)}$$ for real $x>0$ be expressed in relation to the Lambert W Function?

Answer 1

As @Simply Beautiful Art mention in the comments, take square roots of both sides in the equation

$$x=f(x)^2 e^{f(x)} $$

and get

$$\sqrt{x}=f(x) e^{\frac{f(x)}{2}} .$$

Now divide by two both sides

$$\frac{\sqrt{x}}{2}=\frac{f(x)}{2} e^{\frac{f(x)}{2}} $$

Using definition of Lambert W function, we write

$$\frac{f(x)}{2} = W(\pm \sqrt{x}/2) $$

and get

$$f(x) = 2 W(\pm \sqrt{x}/2).$$

As Robert showed in his answer.

Answer 2

$$f(x) = 2 W(\pm \sqrt{x}/2)$$

Answer 3

As written by Robert your function can be written as:

$f(x)=2W({\sqrt{x}\over 2})$

And in general every function of the form:

$x=f^n(x)e^{f(x)}$

Can be written as:

$f(x)=nW({x^{\frac 1n}\over n})$