Q:

$y=e^{-x}$ and $y=x$ point of intersection

How can I find the point of intersection of $y=e^{-x}$ and $y=x$ ? 
Here's the graph

A:

The solution to this equation can be expressed in terms of the Lambert-W function. 
$$ e^{-x} = x $$
$$ 1 = xe^x $$
$$ x = W(1) \approx 0.567$$
Note that the last step is by definition.