Let me add more to your confusion.
y^2 = x -1
"is not a function" if you mean y as a function of x.
It is the square of y, where y is a function of x.
y = f(x) = sqrt[x-1] ------the positive root of the square root only.
If you include the negative root, then y is not f(x) anymore. Because y will have two values for every x-value in the domain.
Domain? Please don't ask me on that.
If that confuses you more, same here.
Heck, who invented functions? And who approved the invention? Those guys can be sued in court.