The function \( y = |x + 5| - 2 \) can be expressed as a piecewise function by considering the definition of the absolute value. The expression \( |x + 5| \) can be defined based on whether the inside expression \( x + 5 \) is non-negative or negative.
- If \( x + 5 \geq 0 \) (or \( x \geq -5 \)), then \( |x + 5| = x + 5 \).
- If \( x + 5 < 0 \) (or \( x < -5 \)), then \( |x + 5| = -(x + 5) = -x - 5 \).
Now, we can write the piecewise definition for the function \( y \):
\[ y = \begin{cases} (x + 5) - 2 & \text{if } x \geq -5 \ (-x - 5) - 2 & \text{if } x < -5 \end{cases} \]
This simplifies to:
\[ y = \begin{cases} x + 3 & \text{if } x \geq -5 \ -x - 7 & \text{if } x < -5 \end{cases} \]
Thus, the correct piecewise definition for \( y = |x + 5| - 2 \) is:
\[ y = \begin{cases} x + 3 & \text{if } x \geq -5 \ -x - 7 & \text{if } x < -5 \end{cases} \]