TITLE: What is the set $g^{-1}([-1,1])$?
QUESTION [5 upvotes]: Let $g(x,y)=xy$ for $xy\in \mathbb R$.
What is $g^{-1}([-1,1])$?
I know that $g^{-1}([-1,1])=\{{(x,y) \,:\    g(x,y)\in [-1,1]}\}$.
But finding all values $(x,y)$ is somewhat confusing.

REPLY [3 votes]: It is confusing. The question is what to do with this set:
$$
\{ (x,y) \mid g(x,y) \in [-1,1]\}
$$
What we want to do is describe what this set is. So we can replace $g(x,y)$ with its definition. Then we want to describe all points $(x,y)$ such that
$$
xy \in [-1,1]
$$
that is, $-1 \le xy \le 1$.
You could stop right here: "it's the set of points $(x,y)$ such that $-1 \le xy \le 1$". But, they probably want you to draw this set.
So we can draw the set by first drawing the points $(x,y)$ where $xy = 1$ and where $xy = -1$.
Then the set of points $(x,y)$ is going to have these lines as a boundary.
So we can draw $y = \frac{1}{x}$, and then draw $y = \frac{-1}{x}$, and then we need to shade the points that are between those two. Try out some points and you will see that anything in-between, such as $(0,0)$, $(0,1)$, $(-1,0)$, $(1/2, -1/2)$ is part of the set where $-1 \le xy \le 1$.