Fixed point of an exponential function

Question

$$f:\Bbb R\to \Bbb R$$ s.t $$f(x)=e^x$$ How many fixed point of f(x) exists?

Answer 1

$$\because \ e^x =x \ \ \ \Rightarrow \frac{1}{e^x}=\frac{1}{x}\\ \\ \\ \therefore x\ e^{-x}=1\ \ \ \Rightarrow -x\ e^{-x}=-1\\ \\ \therefore \ \ z\ e^z \ =-1 \ \ \ \ \therefore \ z=W(-1)\\ where \ W is\ lambert\ function \\ \\$$

and this exist in C not R

so no fixed points for this function\ in this domain