Define a function $f : S_1 \to \mathbb R$ by:
$$ f (\vec x) = (x_1 - 2)^2 - (x_2-1)^2 $$
with domain
$$S_1 = \left\{\vec x \in \mathbb R^2 : 0 \le x_1 \le 1, \lvert x_2\rvert \le \frac{x_1}{2}\right\}.$$
Sketch $S_1$.
How do we go about sketching this?
As mentioned in the comments, demos is a great tool. To graph $S_1$, one can start by considering the inequalities separately. First:
$$0\leq x_1\leq 1.$$
This just determines two parallel vertical lines at $x_1=0$ and $x_1=1$ where our area of interest lies:
Now, if you recall the definition od the absolute value, the second condition that defines $S_1$ is equivalent to:
$$-x_1/2\leq x_2 \leq x_1/2.$$
This gives us two distinct lines between which $S_1$ must lie. They look like:
If we put these two are together, we get that $S_1$ is defined by the intersection:
You can find the boundaries of this triangle, right?
