I've lost count now.

Original link: http://web2.0calc.com/questions/very-hard-math-question-for-my-profile-picture

Hey guys, I've posted this question so many times and still would like an answer. I am just so confused about why it is shaped like that, and what the curve is! Please help me, OK? Thanks!

What I know:

• I graphed it using \(x^{y}=y^{x}\)
• One of the equations is y = x
• The self-intersection point is (e, e)
• The graph is not a hyperbola, it passes through (2, 4) and (4, 2)

What I want to know:

• Why is the intersection point (e, e)?
• What type of graph is this?
• Why does Desmos have such a hard time saying it has a self-intersection point?