If I am provided with function such as this $$f(x) =|x_1|+|x_2|,$$ how do I know what kind of level set it generates? Please, generalize the result. Similarly, for $$f(x) =−(x_1−2)^2+|x_2|+ 1.$$
I don't know how to determine which function generates superlevel, sublevel, or level set.
Hints: Let's look at the level set for 1 for the first function. If $x_1$ and $x_2$ are positive, we can write $x_1+x_2 = 1$. Recognise this as the equation of a straight line and draw this in the positive quadrant of the coordinate system. Now continue for $x_1 < 0, x_2 > 0$ and the other two cases.
The definition of a level set I know is $$ N(f, a) := \{ x \in \mathbb R^2: f(x) \le a\}. $$ Thus the level sets of the first functions are those tilted squares, not just their boundary.
With a similar case distinction to the first function, can you show that for $a \ge 1$, the level set $N(g,a)$ for the second function $g$ consists of two parabolas?