To solve the questions based on the given piece-wise function $y = f(x)$ shown in the image, let's analyze each question in order:

1. Domain of $f$, $D(f)$:

   • The domain is the set of all $x$-values where the function is defined.
   • From the graph, we can see that $f(x)$ is defined from $x = -5$ to $x = 1$ (inclusive of endpoints), and from $x = 1$ to $x = 4$ (not including $x = 1$).
   • Thus, the domain is: $D(f) = [-5,1) \cup (1,4]$

2. Range of $f$, $R(f)$:

   • The range is the set of all $y$-values that the function attains.
   • Observing the graph, the function’s values span from the minimum $y = -6$ up to $y = 6$.
   • So, the range is: $R(f) = [-6,6]$

3. Value of $f(-1)$:

   • To find $f(-1)$, locate $x = -1$ on the graph and find the corresponding $y$-value.
   • At $x = -1$, the function has a value of $y = 2$.
   • Thus, $f(-1) = 2$.

4. Value of $f(2)$:

   • For $f(2)$, observe the graph at $x = 2$.
   • There is an open circle at $x = 2$, indicating that the function is not defined at this point.
   • Therefore, $f(2)$ does not exist (DNE).

Summary of Answers:

1. $D(f) = [-5,1) \cup (1,4]$
2. $R(f) = [-6,6]$
3. $f(-1) = 2$
4. $f(2) = \text{DNE}$

Would you like a further breakdown of each question, or do you have specific inquiries?

Here are some related questions for further exploration:

1. What would be the domain and range if the function had closed intervals throughout?
2. How would the domain change if the function had an open circle at $x = -5$ instead?
3. Can the range include values outside the plotted minimum and maximum if we consider different intervals?
4. How would we denote the domain if the graph extended infinitely?
5. What does the open circle at $x = 2$ signify in terms of continuity?

Tip: Always check for open and closed circles in piece-wise functions to understand continuity and existence of values at specific points.