For the equation $y^y=x^x$, I know that one solution is the line $y=x$ (for $x > 0$), and is shown in this graph here: $y^y=x^x$. However, when I see that graph, I also see a curve that goes from $(0, 1)$ to $1, 0$. Is there an equation (i.e. analytical solution) for just that curve?
I played around with equations, and have discovered that equations in the form $y=\frac{1}{x+a}-a$ kind of fit, but not really. For example, $y=\frac{1}{x+.62}-0.62$ is close, but not really.
I am a high school student and am taking Pre-Calculus, and so my knowledge of advanced functions are limited. However, I do welcome more complicated functions.