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**sarahh** Consider the logistic map $\displaystyle g : [0,1]\implies[0,1], g(x) = rx(1 - x)$, and the sequence of iterates $\displaystyle {x_0}, {x_1} = g({x_0}), ... , {x_{k+1}} = g({x_k})$. Here $\displaystyle r$ is a parameter $\displaystyle 0 < r \le 4$.

So my question is how can I (computationally) show that the map is well defined, that is, the range of g is included in $\displaystyle [0,1]$ for all $\displaystyle 0 < r\le 4$. Also, how can I find the fixed points of $\displaystyle g$ as $\displaystyle 0 < r < 1 $ and $\displaystyle 1 < r < 3$.