TITLE: Why is $\frac{1}{1-x}\circ\frac{1}{1-x}\circ\frac{1}{1-x}=x$ (graphically)?
QUESTION [6 upvotes]: Of course, I can evaluate this composition of functions algebraically which is a fairly elementary exercise. But I'm curious as to whether one can find a (simple) graphical or visual argument (likely based on symmetry, perhaps due to representing this as a composition of the functions $1/x$ and $1-x$). To this extent, I've had a look at the following cobweb plot:

Which represents the composition. But this provides me with no geometric clarity or insight, perhaps because I'm not familiar enough with the geometric properties of hyperbolas? Anyways, if anyone has such an insight or ideas then I'd be grateful.

REPLY [3 votes]: We cube the matrix of your Mobius transformation;
$$
\left(
\begin{array}{cc}
a&b \\
c&d
\end{array}
\right)
$$
means  mapping $$   \frac{ax+b}{cx+d}$$   where we usually write $z$  instead of $x$
$$
\left(
\begin{array}{rr}
0&1 \\
-1&1
\end{array}
\right)
\left(
\begin{array}{rr}
0&1 \\
-1&1
\end{array}
\right)
\left(
\begin{array}{rr}
0&1 \\
-1&1
\end{array}
\right) = 
\left(
\begin{array}{rr}
-1&0 \\
0&-1
\end{array}
\right)
$$
The corresponding Mobius transformation is
$$ \frac{-x}{-1}  = x    $$