Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 5E / iv / Solution

False. Compositions of scalings $z\mapsto \mu z$ and inversion $z\mapsto 1/z$ reduce to one of the forms $\nu z$ or $\nu /z$; they preserve the unordered pair $\{0,\infty \}$. Butis a Möbius transformation fixing $0$ and sends infinity to $1$, so it cannot be such a composition.