To determine the range of the function $f(x)$ based on the graph you uploaded, we need to look at the y-values that the graph covers.

Observations:

• The graph includes several segments and open and closed points.
• For $x < -4$, the function appears to be constant at $f(x) = 3$ but with an open circle at $(-4, 3)$, meaning $f(x) = 3$ is not included at that point.
• At $x = -4$, $f(x) = 1$, as indicated by the closed point.
• From $x = -4$ to $x = -3$, the function decreases linearly, going from $f(x) = 1$ to $f(x) = 0$, where there is an open circle at $(-3, 0)$, so $f(x) = 0$ is not included at $x = -3$.
• At $x = -3$, $f(x) = 0.5$, as indicated by the filled circle.
• From $x = -3$ to $x = 0$, the graph continues to drop down to $f(x) = -2$, with a closed point at $(0, -1)$.
• After $x = 0$, the graph curves upwards and rises to $f(x) = 4$ as $x$ increases.

Range of the function:

• The function attains values from $-2$ (the lowest point) to 4 (the highest point).
• Specifically, the range includes the y-values:
  • $1$ (because of the closed circle at $(-4, 1)$),
  • All values between $-2$ and $4$, but excludes 0 (because of the open circle at $(-3, 0)$).

Conclusion:

The range of the function is:

[ \boxed{[-2, 1] \cup (1, 4]} ]

This accounts for all the y-values the function can take based on the graph.

Would you like further details or have any specific questions?

Here are some related questions to expand on this topic:

1. What is the difference between open and closed intervals in function ranges?
2. How does one determine the domain from a graph of a function?
3. Can a function have gaps or breaks in its range? What does that mean?
4. How would the range change if a point on the graph were filled instead of open?
5. What are some real-life examples of functions with restricted ranges?

Tip: Always pay attention to open and closed circles when determining the range from a graph, as they indicate whether endpoints are included or excluded.