The problem provided asks you to analyze the absolute value function $f(x) = |x + 3| + 2$, identify its domain and range, and choose the correct graph.

Step-by-Step Breakdown:

1. The given function: $f(x) = |x + 3| + 2$ This is an absolute value function. The graph of $f(x) = |x + 3|$ is a V-shaped graph that shifts horizontally 3 units to the left (since $x + 3$) and vertically 2 units upward (because of the +2).

2. Domain: Absolute value functions have no restrictions on $x$. Therefore, the domain is all real numbers: $\text{Domain}: (-\infty, \infty)$

3. Range: The range is determined by the vertical shift. The lowest value of $|x + 3|$ is 0, and adding 2 shifts this minimum value up to 2. Therefore, the function will only output values greater than or equal to 2: $\text{Range}: [2, \infty)$

4. Graph: The graph of the function $f(x) = |x + 3| + 2$ is a V-shape with its vertex at $(-3, 2)$. The correct graph should reflect this vertex and the upward-facing V shape.

Identifying the correct graph:

Based on the graph choices:

• The correct graph should have a vertex at $(-3, 2)$.
• Looking at the image, Graph B is the one with the vertex at $(-3, 2)$.

Final answers:

• Domain: $(-\infty, \infty)$
• Range: $[2, \infty)$
• Correct graph: B

Would you like any further details on how these conclusions were reached?

Relative Questions:

1. How do horizontal and vertical shifts affect the graph of an absolute value function?
2. Why does the domain of an absolute value function always span all real numbers?
3. How do we determine the range of absolute value functions?
4. How can the vertex of an absolute value function be found from its equation?
5. What are the general properties of absolute value graphs in terms of shape and transformations?

Tip: Always check the vertex of the absolute value function to verify which graph is correct based on shifts and transformations.