To determine the correct inequalities for the domain and range of the linear function $g$ based on its graph, let's first clarify what domain and range represent:

1. Domain: The set of all possible $x$-values covered by the graph (the horizontal span).
2. Range: The set of all possible $g(x)$-values, which are the same as the $y$-values (the vertical span).

Assuming that we have the portion of the graph on hand, here's how we analyze it:

• Domain: The domain refers to the minimum and maximum $x$-values shown on the graph.
• Range: The range refers to the minimum and maximum $g(x)$-values (or $y$-values).

Given Options:

Let's evaluate the choices one by one.

A. Domain: $-7 < x < 6$, Range: $-4 < g(x) < 5$
This indicates that the graph covers $x$-values from $-7$ to $6$, but not including these endpoints, and $y$-values from $-4$ to $5$, also excluding the endpoints.

B. Domain: $-7 \leq x \leq 6$, Range: $-4 \leq g(x) \leq 5$
This indicates that the graph covers $x$-values from $-7$ to $6$, including these endpoints, and $y$-values from $-4$ to $5$, including the endpoints as well.

C. Domain: $-4 \leq x \leq 5$, Range: $-7 \leq g(x) \leq 6$
This suggests that $x$-values are from $-4$ to $5$, and $y$-values are from $-7$ to $6$, both endpoints included.

D. Domain: $-4 < x < 5$, Range: $-7 < g(x) < 6$
This suggests $x$-values are from $-4$ to $5$, but endpoints are not included, and $y$-values from $-7$ to $6$, also excluding endpoints.

To choose the best answer:

• If the graph shows solid dots at the endpoints, it indicates that the endpoints are included, meaning the domain and range should have "≤" rather than strict inequality "<".
• If the graph shows open dots, the endpoints are not included, which would require strict inequalities ("<").

Conclusion:

If the graph shows closed dots at $x = -7$ and $x = 6$ and similar for the $y$-values, the correct answer is B.

Otherwise, if the endpoints are not included, A would be correct.

Do you need more details or have any further questions?

Related Questions:

1. How do you determine the domain from a graph?
2. How can you distinguish between closed and open intervals?
3. What is the difference between $g(x)$ and $f(x)$ in linear functions?
4. How do inequalities affect the graph of a linear function?
5. What is the impact of endpoint inclusion on real-world applications?

Tip: Always look for open or closed dots at the endpoints of a graph to decide whether the inequality includes the boundary points.