What numbers can be created by $1-x^2$ and $x/2$? Suppose I have two functions
$$f(x)=1-x^2$$
$$g(x)=\frac{x}{2}$$
and the number $1$. If I am allowed to compose these functions as many times as I like and in any order, what numbers can I get to if I must take $1$ as the input? For example, I can obtain $15/16$ by using
$$(f\circ g\circ g)(1)=\frac{15}{16}$$
It is obvious that all obtainable numbers are in the set $\mathbb Q\cap [0,1]$, but some numbers in this set are not obtainable, like $5/8$ (which can be easily verified).
Can someone identify a set of all obtainable numbers, or at least a better restriction than $\mathbb Q\cap[0,1]$? Or, perhaps, a very general class of numbers which are obtainable?
 A: Not really an answer, but something possibly important and interesting.
Suppose we consider this problem as a random dynamical system starting with the seed $1/2$ and applying either the function $f$ or the function $g$ with equal likelihood at each step, repeating this process ad infinitum. Let us define the function $d(x)$ as the asymptotic density of points less than $x$; that is, the limit of the proportion of points less than $x$ as the number of iterations approaches infinity. Then one may establish the following functional equation using a probabilistic argument:
$$d(x)=\frac{1}{2}\cdot d(2x)+\frac{1}{2}\cdot (1-d(\sqrt{1-x}))$$
or
$$d(x)=\frac{d(2x)-d(\sqrt{1-x})+1}{2}$$
Now observe that the number $m=\frac{\sqrt{17}-1}{8}$ satisfies $2m=\sqrt{1-m}$, implying that
$$d\bigg(\frac{\sqrt{17}-1}{8}\bigg)=\frac{1}{2}$$
and so approximately half of the points will be less than $m=\frac{\sqrt{17}-1}{8}$ for a large number of iterations (we've basically found the median of our data set).
By using the fact that $d(x)=d(1/2)\space\forall x\in [1/2,3/4]$ and the fact that $d(x)=1\space\forall x\ge 1$, one may also calculate the following special values:
$$d\bigg(\frac{1}{2}\bigg)=\frac{2}{3}$$
$$d\bigg(\frac{\sqrt{17}+23}{32}\bigg)=\frac{3}{4}$$
$$d\bigg(\frac{239-23\sqrt{17}}{512}\bigg)=\frac{3}{8}$$
A: Here's a sorted plot of all distinct values of compositions of up to 22 elementary functions f and g:

Mathematica code:
f[x_] := 1 - x^2;
g[x_] := x/2;
DeleteDuplicates[
  Sort[
         (Apply[Composition, #][x] /. x -> 1/2) & /@ 
         Flatten[Table[Tuples[{f, g}, i], {i, 22}], 1]]]

ListPlot[%]

This graph confirms the obvious fact that the value can never be greater than $1$ and that there is a gap between $1/2$ and $3/4$.
