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 $y \geq 6$."

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.