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Let's analyze the problem step by step.

### 1. Identifying the graph of \( g(x) = |x| + 6 \):
- The function \( g(x) = |x| + 6 \) is a transformation of the basic absolute value function \( f(x) = |x| \). The transformation involves a **vertical shift** of 6 units upwards, as the "+6" translates the graph upwards by 6 units.
  
- Looking at the graphs, the correct graph must have a "V" shape (since it is an absolute value function), and its vertex must be at \( (0, 6) \).

By this reasoning, the **third graph** (bottom-left) is the correct one because the vertex is at \( (0, 6) \).

### 2. Description of the transformation:
- The transformation from \( f(x) = |x| \) to \( g(x) = |x| + 6 \) is a **vertical translation 6 units up**, as the "+6" shifts the graph upward.

Therefore, the correct choice is: **"The graph of \( g \) is a vertical translation 6 units up of the graph of \( f \)."**

### 3. Domain and range:
- The domain of \( g(x) = |x| + 6 \) is the same as the domain of the absolute value function, which is **all real numbers** because the absolute value function is defined for every real number \( x \).

- The range of \( g(x) = |x| + 6 \) is \( y \geq 6 \) because the lowest value of \( |x| \) is 0, and adding 6 shifts the minimum value of the function to 6.

Thus, the correct choice is: **"The domain is all real numbers. The range is \( y \geq 6 \)."**

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Would you like further clarification or more details? Here are some additional questions you might find useful:

1. What are the properties of the absolute value function \( |x| \)?
2. How do vertical and horizontal translations affect the graph of a function?
3. Can you explain how the domain and range of absolute value functions are determined?
4. What are other transformations (besides vertical shifts) that could be applied to \( |x| \)?
5. How can you graph \( g(x) = |x| + k \) for different values of \( k \)?

### Tip: 
Always pay attention to whether transformations involve vertical or horizontal shifts. Vertical shifts are controlled by adding/subtracting outside the function, while horizontal shifts are inside.

Identify the graph of g(x) = |x| + 6 and choose the correct description of the transformation from the graph of f(x) = |x| to the graph of g. Choose the description of the domain and range.

Learn how to identify and graph the absolute value function g(x) = |x| + 6, understand vertical translations, and determine the domain and range of the function. This problem covers graph transformations of absolute value functions, suitable for students in Grades 9-10.

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