I understood that a Mobius transformation can be decomposed into a composition of elementary operations, e.g., rotation, dilation, and inversion.
Here, I'm facing the following transformation to understand; $f(z) = \frac{z + 2}{3z + 4}$. I could factorise it till the following but was not sure how to understand the remaining factor. $f(z) = f_1 \circ f_2 \circ f_3 \circ f_4$
$$ f_4: z \mapsto 3z, f_3: z \mapsto z + 4, f_2: z \mapsto 1/z, f_1: z \mapsto z * z; $$.
Then I got the remaining term of $\frac{+2}{3z + 4}$. How can I deal with this?