How I can find a similar expression for $x₀>1/2$

Question

For the logistic map http://mathworld.wolfram.com/LogisticMapR=2.html the formula (4) in the link is valid only for $x₀<1/2$. How I can find a similar expression for $x₀>1/2$. The same question for $r=-2$ (defined only in an interval of lenght $0.5$). See here http://mathworld.wolfram.com/LogisticMapR=-2.html

Answer 1

Since the expression $rx(1-x)$ is symmetric under $x\to1-x$, it's OK here to use absolute value signs in the logarithm:

$$x_n={1\over2}\left(1-\exp(2^n\ln\left|1-2x_0\right|)\right)$$

This now works for all $x_0$. (One caveat: when $n=0$, the expression on the right hand side returns the smaller of $x_0$ and $1-x_0$.)