To solve this problem, we need to determine the range of the linear function $g(x)$, based on the graph shown.

Key steps:

1. Examine the vertical axis (y-axis): This axis gives the values of $g(x)$, which represent the function's output.
2. Identify the minimum and maximum values of $g(x)$: The lowest and highest points on the line segment shown in the graph represent the range.

From the image, it appears that:

• The lowest value of $g(x)$ is $-3$, as the graph touches the point at $y = -3$.
• The highest value of $g(x)$ is $4$, as the graph reaches up to $y = 4$.

Thus, the range of the function $g(x)$ is between $-3$ and $4$, inclusive of $4$, but not including $-3$ since the line does not touch the exact value.

Correct inequality:

The best inequality that describes this range is: $-3 < g(x) \leq 4$

Thus, the correct answer is the third option: Range: $-3 < g(x) \leq 4$.

Do you have any questions or would you like more details on the process?

Here are 5 related questions to expand on this:

1. How do we define the domain of a function?
2. What is the difference between the range and the domain?
3. How would the graph change if the function had a range $-4 < g(x) \leq 5$?
4. What does it mean for a function to have an "open" vs. "closed" interval for its range?
5. How can we algebraically calculate the range of a linear function from its equation?

Tip: Always check whether the endpoints are included in the range or domain by observing if the graph touches those points exactly.