Theorem: LFF.COMP:
For any linear fractional function,$f(x) =\frac {ax+b}{cx+d}$
with $c \ne 0$,
there are numbers $A, h$, and $k$, so that $f$
can be expressed as a composition of the core negative
power function, $R(x) =\frac 1 x$, with
core linear functions:
$f(x) = (f_{+k} \circ f_{*A} \circ
R \circ f_{-h} )(x)= \frac A {x-h} + k$
where $f_{+k}(u)=u+k $, $f_{-h}(u)=u-h $, and $
f_{*A}(x)=Ax$.