**Step 1: Case $c \neq 0$.** Let $f(z) = \frac{az + b}{cz + d}$. Perform polynomial long division on the numerator: \begin{align*} f(z) = \frac{a}{c} + \frac{b - ad/c}{cz + d} = \frac{a}{c} + \frac{bc - ad}{c^2} \cdot \frac{1}{z + d/c}. \end{align*} This can be realised as the composition: \begin{align*} z \xrightarrow{z + d/c} z + \frac{d}{c} \xrightarrow{1/w} \frac{1}{z + d/c} \xrightarrow{\alpha w} \frac{bc - ad}{c^2(z + d/c)} \xrightarrow{w + a/c} \frac{a}{c} + \frac{bc - ad}{c^2(z + d/c)} = f(z), \end{align*} where $\alpha = (bc - ad)/c^2 \neq 0$. Each arrow is a translation, inversion, or dilation/rotation. **Step 2: Case $c = 0$.** Then $f(z) = \frac{az + b}{d} = \frac{a}{d}z + \frac{b}{d}$, which is the composition of the dilation $z \mapsto \frac{a}{d}z$ followed by the translation $z \mapsto z + \frac{b}{d}$.