To solve the problem of finding the domain and range of the given function $f$ based on the graph, let’s carefully analyze it step by step:

1. Domain of the Function:

The domain refers to all possible $x$-values for which the function $f$ is defined.

From the graph:

• The function is not defined at $x = -2$ (there is a hole at this point).
• For all other $x$-values, the graph exists.

Thus, the domain is: $(-\infty, -2) \cup (-2, \infty)$

2. Range of the Function:

The range refers to all possible $y$-values that the function can take.

From the graph:

• The function is not defined at $y = -2$ (corresponding to the hole).
• The function does not include the value $y = -1$ (this is visible because of the jump in the graph).

Thus, the range is: $(-\infty, -2) \cup (-1, \infty)$

3. Correct Answer:

The correct choice is D:

• Domain: $(-\infty, -2) \cup (-2, \infty)$
• Range: $(-\infty, -2) \cup (-1, \infty)$

Do you need further explanation or clarification? Let me know!

Here are 5 related questions for practice:

1. What is the domain and range of a horizontal line $y = 3$?
2. How do you find the domain and range of a function with a hole?
3. Explain the concept of intervals and union notation in mathematics.
4. Determine the domain and range for the function $f(x) = \frac{1}{x+3}$.
5. How does a vertical asymptote affect the domain of a function?

Tip: To identify the domain and range of a graph, look for breaks (holes, asymptotes, or jumps) in the graph!