https://file.aiquickdraw.com/mathbot/file/1725571124737umgjo4w1.jpg

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.

g(x) = (1/2) |x + 1| - 4. Determine the domain (D) and range (R) of the function.

Learn how to graph the absolute value function g(x) = (1/2)|x + 1| - 4 and determine its domain and range. This step-by-step guide explains key concepts of graph transformations and absolute value functions. Suitable for Grades 9-11.

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