Let $S=\{(x,y):x^2+y^2=1,-1\le x\in \mathbb{R}\le 1\text{ and }-1\le y\in \mathbb{R}\le 1\}$. Which one of the following is correct?

The equation $x^2+y^2=1$ represents a circle with center (0,0) and radius 1.For it to be a function, each input $x$ must have exactly one output $y$.For $x=0$, $y^2=1⟹y=\pm 1$.Since one input ($x=0$) corresponds to two outputs ($y=1$ and $y=-1$), $S$ fails the vertical line test and is not a function.