From the description of the graph, let's break down the key observations:

1. First part of the graph:

• The curve is linear and enters at the point $(-3, 0)$, increases as it moves to the right, and ends at the open point $(-1, 2)$.
  • Since it’s a straight line and the function is defined everywhere along this segment, we know that the function is continuous over this section of the domain.

2. Second part of the graph:

• This part starts from the closed point $(-1, 1)$, suggesting that the function is defined at $x = -1$ with value 1.
• The curve then moves downward and right, crossing the x-axis at $x = -0.5$, and the y-axis at $y = -0.75$.
• The function changes direction at $(0.5, -1.9)$, then moves upward, crossing the x-axis at $x = 1.2$, and exits in the first quadrant.

Continuity at $x = -1$:

The key issue is at $x = -1$, where the function transitions between the two parts:

• Left-hand limit (as $x \to -1^-$): From the description of the first part, the function approaches $(-1, 2)$ from the left, which means: $\lim_{x \to -1^-} f(x) = 2$

• Right-hand limit (as $x \to -1^+$): From the description of the second part, the function starts at the closed point $(-1, 1)$, so: $\lim_{x \to -1^+} f(x) = 1$

Since the left-hand limit $\lim_{x \to -1^-} f(x) = 2$ is not equal to the right-hand limit $\lim_{x \to -1^+} f(x) = 1$, the limit at $x = -1$ does not exist. Therefore, the function is not continuous at $x = -1$.

Conclusion:

The function is not continuous at $x = -1$ because: $\lim_{x \to -1^-} f(x) \neq \lim_{x \to -1^+} f(x)$ Thus, the function is not continuous on its domain.

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Would you like further details or an explanation on how to handle limits or continuity at specific points?

Here are some follow-up questions to dive deeper:

1. What is the definition of continuity for a function?
2. How do you evaluate limits from the left and right of a point?
3. What is the difference between open and closed points on a graph?
4. How do you determine if a function is continuous at a specific point or interval?
5. What happens to the continuity of a function if the limits at a point do not match?

Tip: To check for continuity at a point, always ensure that the left-hand limit, right-hand limit, and the function's value at the point all match.