Deeper meaning and intuition behind $\frac{x}{1+x^2}$ having the same values for $x$ and for $\frac{1}{x}$ I was playing around with the function $f(x)=\frac{x}{1+x^2}$ and I noticed that $f(x)$ has the same values for $x$ and for $\frac{1}{x}$. 
Is there any intuition or deeper meaning behind this?
Thanks in advance!

Consider $g(x) = x + \frac{1}{x}$. Then $g(x) = g(\frac{1}{x})$, because $x + \frac{1}{x}$ = $\frac{1}{x} + \frac{1}{\frac{1}{x}} = \frac{1}{x} + x$.
Your function $f(x) = \frac{1}{g(x)} = \frac{x}{x} \frac{1}{x + \frac{1}{x}} = \frac{x}{x(x + \frac{1}{x})} = \frac{x}{x^2 + 1}$
So some intuitions or deeper meanings behind this are: addition commutes and a reciprocal of a reciprocal is the original value (inverse of inverse). BTW note that $x\ne0$ for this to work - which we know it is, because we have already $\frac{1}{x}$.
