Question

Daca x ∈ [-1; 0] si -2y ∈ [-2; 2], atunci (x-2y)² apartine intervalului...?

explicati daca se poate

Answer 1

$-1 \leq x \leq 0 \\ -2 \leq -2y \leq 2 \\ -------(+) \\ \\ -1+(-2) \leq x + (-2y) \leq 0+2\\ -3 \leq x-2y \leq 2 \\ \\ \Rightarrow x-2y \in [-3,2] \\ \\ Stim ca: \\ \\ a^2 = (|a|)^2, intotdeauna. \\ \\ x-2y \in [-3,2]\Big| |\square| \Rightarrow\\ \\ \Rightarrow |x-2y| \in \Big([-3,2] \backslash[-3,0)\Big) \cup (0,3]\\ \\ Stim ca aflarea modulului dintr-o \\ multime implica extragerea tuturor \\ numerelor negative din acea multime si\\ adaugarea lor in multime ca numere \\ pozitive. \\ \\ \Rightarrow |x-2y| \in [0,3] \Big|^2 \\ \\ \Rightarrow \Big(|x-2y|\Big)^2\in [0^2,3^2]\\ \\ \Rightarrow \Big(|x-2y|\Big)^2\in [0,9]\\ \\ \Rightarrow (x-2y)^2 \in [0,9]$