I would like to apply the following rule in order to find the inverse function of a given function :
if function $f(x)= h(i(j(x)))$ and if functions $h(x)$, $i(x)$ and $j(x)$ are invertible, then $f^{-1}(x) = j^{-1}(i^{-1}(h^{-1}(x)))$.
In order to apply this rule to a rational function such as :
$ f(x) = \frac {6x+4} {4x+5}$ ( Source : Larson's College Algebra, Unit 2.7, exercise #53)
I would need to express function $f$ as a composition of elementary invertible functions.
Is this possible?
Hence my question: suppose a rational function $f$ is a quotient of two first degree polynomials, is it always possible to express $f$ as a composition of elementary ( invertible) functions?
Think of this as a Möbius Transformation. I'm sure this https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Simple_M%C3%B6bius_transformations_and_composition answers your question positively. It can always be done.