Hey math mates! I have a (pretty dumb) question about inequalities, I'm preparing an partial exam for calculus and I'm struggling with this:

I have to find the domain of F(x,y)= arcsin ([tex]\frac{y}{x}[/tex]). These are basic exercises of one of the first topics, like an introduction to multivariable calculus.

Since the arcsin function is bounded between [-1;1], I put the restriction [tex]-1\le (\frac{y}{x}) \le 1[/tex] to the argument. Then, solving that, it finally is [tex]-x \le y \le +x[/tex]. They ask you to trace a graph and I find something like this:

But it has to be like this:

I just cannot realize how, analytically, deduce that the other side of the graph (corresponding to the reversed inequation: [tex]-x \ge y \ge +x[/tex]) also belongs to the domain. The book says that is either one or another of the possibilities, depending to the value of [tex]x[/tex] (negative or positive). Isn't it just implied in [tex]-x \le y \le +x[/tex] that the input can be [tex]x[/tex] positive or negative?

Because if you pull apart the double inequation [tex]-x \le y \le +x[/tex] into [tex]-x \le y[/tex] and [tex]y \le +x[/tex] it's also the same, isn't it? Just can't figure it out... i'm done. Can I be so dumb?

Cheers, Mark.