https://file.aiquickdraw.com/mathbot/file/1726687819904s6sijn1n.jpg

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.

A part of linear function g is graphed on the grid. Which inequalities best describe the range of the part shown?

This problem focuses on determining the range of a linear function from its graphical representation. It covers key concepts such as graphing inequalities and analyzing the range of functions, suitable for students in Grades 8-10.

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