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.