Question

f:R-R, f(x)=$x^{2}$-2x, d: y=2x-3, aflați $G_{f}$∩d.

Answer 1

 

$\displaystyle\bf\\ f(x)=x^2-2x~~~este~o~parabola\\ ~~~~~y=2x-3~~~~este o dreapta\\O~parabbola~si~o~dreapta~pot~avea:\\2~puncte~de~intersectie~daca~dreapta~este~o~secanta~a~parabolei.\\1~puncte~de~intersectie~daca~dreapta~este~o~tangenta~a~parabolei.\\0~puncte~de~intersectie~daca~dreapta~si~parabola~nu~se~intalnesc.$

.

$\displaystyle\bf\\Aflam~coordonata~x~a~punctelor~de~de~intersectie~din~egalitatea:\\\\f(x)=y\\x^2-2x=2x-3\\x^2-2x-2x+3=0\\x^2-4x+3=0\\x^2-x-3x+3=0\\x(x-1)-3(x-1)=0\\(x-1)(x-3)=0\\\boxed{\bf x_1=1}\\\boxed{\bf x_2=3}\\\\Aflam~coordonata~y~a~punctelor~de~intersectie~din~ecuatia~dreptei:\\y_1=2x_1-3=2\cdot1-3=2-3=-1\\\\\boxed{\bf y_1=-1}\\\\y_2=2x_2-3=2\cdot3-3=6-3=3\\\\\boxed{\bf y_2=3}\\\\G_f\bigcap d=A(x_1,~y_1)~~si~~B(x_2,~y_2)\\\\\boxed{\bf G_f\bigcap d=A(1,~-1)~~si~~B(3,~3)}$