https://file.aiquickdraw.com/mathbot/file/17291715105361c2g4o1n.jpg

Let's analyze the function \( y = |x + 3| - 2 \) and determine its domain and range in set-builder and interval notations.

### Step 1: Domain
The function \( y = |x + 3| - 2 \) involves an absolute value, which means it is defined for all real numbers because the absolute value function has no restrictions. Thus, the **domain** is all real numbers:

- In set-builder notation: \( \{ x \mid x \in \mathbb{R} \} \)
- In interval notation: \( (-\infty, \infty) \)

### Step 2: Range
To determine the range, consider how the function behaves.

1. The smallest value that the absolute value \( |x + 3| \) can take is 0 (when \( x = -3 \)).
2. When \( x = -3 \), \( y = |x + 3| - 2 = 0 - 2 = -2 \).
3. As \( x \) moves away from -3, \( |x + 3| \) increases, so \( y \) increases from -2 upwards.

Thus, the **range** of the function starts at \( -2 \) and goes to infinity:

- In set-builder notation: \( \{ y \mid y \geq -2 \} \)
- In interval notation: \( [-2, \infty) \)

### Conclusion:
The correct option from the image should be:
- **Domain**: \( \{ x \mid x \in \mathbb{R} \}; (-\infty, \infty) \)
- **Range**: \( \{ y \mid y \geq -2 \}; [-2, \infty) \)

This matches the third option in the image.

Would you like more details or have any questions? Here are 5 related questions to expand on this:

1. How do absolute value functions typically affect the range of a function?
2. Why is the domain of a function involving absolute values usually all real numbers?
3. How does shifting the graph of an absolute value function affect its domain and range?
4. Can a function involving absolute value ever have a negative range?
5. What are some common transformations that can affect the range of a function?

**Tip**: For any function involving absolute values, the range is influenced by how the function shifts vertically or stretches, but the domain usually remains all real numbers unless specified otherwise.

What are the domain and range of the function defined by y = |x + 3| - 2 written in set-builder notation and interval notation?

Understand how to find the domain and range of the function y = |x + 3| - 2 using set-builder and interval notation. This example covers absolute value properties, showing how shifts affect the function's range. Suitable for high school students in Grades 9-12.

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