# 2D graph of an inequality (of a 3 variable function domain)

### 2D graph of an inequality (of a 3 variable function domain)

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 ($$\frac{y}{x}$$). 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 $$-1\le (\frac{y}{x}) \le 1$$ to the argument. Then, solving that, it finally is $$-x \le y \le +x$$. 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: $$-x \ge y \ge +x$$) also belongs to the domain. The book says that is either one or another of the possibilities, depending to the value of $$x$$ (negative or positive). Isn't it just implied in $$-x \le y \le +x$$ that the input can be $$x$$ positive or negative?

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

Cheers, Mark.
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