I have two relations:
$$M_1 = (x,y)\qquad x²+y²+6y = 7 $$
$$M_2 = (x,y)\qquad x²+y²-6x = 7, \qquad y \ge 0$$
The question is if this relations also reflex functions like $x \rightarrow y$?
I think I have to test if I also can write the relations like $x \rightarrow y$ right?
So what I did was:
$$M_1 = (x,y)\qquad x^2+y^2+6y = 7 $$
$$M_1 = (x,y)\qquad x= \sqrt{-y^2-6y+7} $$
My question us can i short it better? And is $M_1$ now a function like $x \rightarrow y$? I would say that like i have it now it wouldnt be the same like $x \rightarrow y$ because of the $y^2$?
$$M_1:x^2+y^2+6y=7\iff x^2+y^2+2\cdot3y+3^2=7+3^2\iff x^2+(y+3)^2=16=4^2$$
$\iff M_1$ is a circle centered in $(0,-3)$ with radius $4$. Also, $y(x)=-3\pm\sqrt{16-x^2}$.
$$M_2:x^2+y^2-6y=7\iff x^2+y^2-2\cdot3y+3^2=7+3^2\iff x^2+(y-3)^2=16=4^2$$
$\iff M_2$ is a circle centered in $(0,3)$ with radius $4$. Also, $y(x)=3\pm\sqrt{16-x^2}$.