To analyze the graph you've provided, let's break it down step by step:

1. Determining if it's a Function:

A graph represents a function if, for every $x$-value, there is at most one $y$-value. This is commonly tested using the Vertical Line Test. If any vertical line drawn across the graph intersects the graph in more than one point, then the graph does not represent a function.

In this graph, each vertical line intersects the graph at most once. Therefore, this graph represents a function.

2. Finding the Domain:

The domain consists of all the possible $x$-values that the function can take.

• From the graph, the function starts at $x = -4$ (where the circle is open, indicating that -4 is not included) and ends at $x = 3$ (where the circle is closed, indicating that 3 is included).
• So, the domain is $\left( -4, 3 \right]$.

3. Finding the Range:

The range consists of all the possible $y$-values that the function can output.

• The $y$-values span from $y = -3$ (where the circle is closed, indicating that -3 is included) to $y = 2$ (where the circle is closed, indicating that 2 is included).
• So, the range is $\left[ -3, 2 \right]$.

Summary:

• Function: Yes, it is a function.
• Domain: $\left( -4, 3 \right]$
• Range: $\left[ -3, 2 \right]$

Would you like any additional details or explanations on any part of this analysis?

Follow-up Questions:

1. What happens to the domain if the graph includes or excludes endpoints differently?
2. How does the range change if the graph shifts vertically?
3. Can you explain the significance of open and closed circles in determining the domain and range?
4. How would you determine the domain and range for more complex graphs?
5. What is the importance of understanding the domain and range when dealing with real-world functions?

Tip: Always use the Vertical Line Test to check if a graph represents a function. It’s a quick and reliable method.