How to take composition of 2 bilinear transforms

Question

I want to take the composition of the following 2 bilinear tranforms:

1) $z\mapsto z/i$

2) $z\mapsto -2z/ ((1+i)z+2i)$

How do I do this?

Answer 1

$$f_2\bigl(f_1(z)\bigr)=\frac{-2\frac zi}{(1+i)\frac zi +2i}.$$

You may consider a function as an action. So $f_1$ is described by the action: “Divide by $i$!” Hence $f_1\bigl(f_2(z)\bigr)$ calls you: “Divide $f_2(z)$ by $i$!” Now do so, please.