The problem provided shows the function:

$g(x) = \frac{1}{2} |x + 1| - 4$

and a coordinate grid for graphing it. Let's analyze the function step by step to understand how to graph it and determine its domain (D) and range (R).

Step-by-step Analysis:

1. Understanding the Function:
   The function $g(x) = \frac{1}{2} |x + 1| - 4$ is an absolute value function, which means it has a "V" shape. The basic form of an absolute value function is $y = a|x - h| + k$, where $(h, k)$ is the vertex of the "V".

2. Vertex of the Function:
   For $g(x) = \frac{1}{2} |x + 1| - 4$, we can rewrite the inside as $|x - (-1)|$. Therefore, the vertex of the function is at $(-1, -4)$.

3. Slope of the Function:
   The coefficient $\frac{1}{2}$ outside the absolute value indicates a vertical compression by a factor of $\frac{1}{2}$. This affects the slope of the lines to the left and right of the vertex.

4. Graphing the Function:

   • Start by plotting the vertex at $(-1, -4)$.
   • For $x > -1$, the function behaves like $g(x) = \frac{1}{2}(x + 1) - 4$. The slope is $\frac{1}{2}$.
   • For $x < -1$, the function behaves like $g(x) = \frac{1}{2}(-(x + 1)) - 4 = -\frac{1}{2}(x + 1) - 4$. The slope is $-\frac{1}{2}$.

5. Domain and Range:

   • Domain (D): The domain of the function is all real numbers because there are no restrictions on $x$. Thus, $D: (-\infty, \infty)$.
   • Range (R): The lowest point on the graph is at the vertex $(-1, -4)$. Since the function increases in both directions from the vertex, the range is $R: [-4, \infty)$.

Would you like a detailed plot of the graph or have any questions about the steps?

Relative Questions:

1. How do transformations affect the graph of an absolute value function?
2. What is the effect of changing the coefficient of $x$ inside the absolute value function?
3. How would the graph change if the function were $g(x) = |x + 1| - 4$ instead?
4. What is the significance of the domain and range in the context of real-world applications?
5. How do you find the axis of symmetry for absolute value functions?

Tip:

When working with absolute value functions, always identify the vertex first, as it is the key point around which the graph is shaped.