Question

Pe multimea numerelor reale se defineste legea de compozitie x*y=2xy-2x+3
a) arati 2020*1=1
b) demonstrati ca x*y=2(x-1)(y-1)+1
C) determinati nr reale x pentru care (x*x)*x=x

Answer 1

$\bf Pe \ mul\c{\bf t}imea\ numerelor\ reale\ se \ d efine\d s te\ legea\ de\ compozi\c{\bf t}ie:\\ \\ \it x*y=2xy-2x-2y+3\\ \\ \bf\ a)\ ar\breve ata\c{\bf t}i\ c\breve a\ \ \it 2020*1=1\\ \\ \bf b)\ demonstra\c{\bf t}i\ \ c\breve a\ \ \ \it x*y=2(x-1)(y-1)+1\\ \\ \bf c)\ determina\c{\bf t}i\ \ x\in\mathbb{R},\ \ pentru\ \ care\ \ \it (x*x)*x=1$

Rezolvare:

$\it b)\ \ 2xy-2x-2y+3= 2xy-2x-2y+2+1=2(xy-y-x+1)+1=\\ \\ =2[y(x-1)-(x-1)]+1=2(x-1)(y-1)+1$

$\it a)\ 2020*1=2(2020-1)(1-1)+1=1$

$\it c)\ \ (x*x)*x+1\ \Leftrightarrow\ [2(x-1)(x-1)+1]*x=1 \Leftrightarrow\\ \\ \Leftrightarrow\ [2(x-1)^2+1]*x=1 \Leftrightarrow 2\{[2(x-1)^2+1-1](x-1)\}+1=1 \Leftrightarrow\\ \\ \Leftrightarrow \ 4(x-1)^3=0 \Leftrightarrow x-1=0 \Leftrightarrow x=1$