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  $f_{*A}(x)=Ax$.