Question Number 221154 by gregori last updated on 25/May/25
$$\:{f}\left({x}\right)=\:\frac{{x}}{\mid\:{x}\:\mid\:+\:\mathrm{1}} \\ $$$$\:\:{f}\left({f}\left({f}\left({f}\left({x}\right)\right)\right)\right)\:=? \\ $$
Commented by Frix last updated on 25/May/25
$$\frac{{x}}{\mathrm{4}\mid{x}\mid+\mathrm{1}} \\ $$$${f}_{\mathrm{1}} \left({x}\right)=\frac{{x}}{\mid{x}\mid+\mathrm{1}} \\ $$$$\forall{n}\in\mathbb{N}\wedge{n}>\mathrm{1}:\:{f}_{{n}} \left({x}\right)={f}_{\mathrm{1}} \left({f}_{{n}−\mathrm{1}} \left({x}\right)\right)=\frac{{x}}{{n}\mid{x}\mid+\mathrm{1}} \\ $$