To convert the function $f(x) = -|x + 4|$ into a piecewise function, we need to consider the behavior of the absolute value expression $|x + 4|$. The absolute value causes the function to have different expressions depending on whether the input expression inside the absolute value is positive or negative.

1. Identify critical point: The absolute value changes expression at the point where $x + 4 = 0$. Solving for $x$, we get $x = -4$.

2. Define expressions on each interval:

   • When $x + 4 \geq 0$ (i.e., $x \geq -4$), the expression inside the absolute value is non-negative. So, $|x + 4| = x + 4$.
   • When $x + 4 < 0$ (i.e., $x < -4$), the expression inside the absolute value is negative. Thus, $|x + 4| = -(x + 4) = -x - 4$.

3. Piecewise function: We can now write $f(x)$ as:

   - (x + 4) & \text{if } x \geq -4 \\ - (-x - 4) & \text{if } x < -4 \end{cases}$$ Simplifying each case: $$f(x) = \begin{cases} -x - 4 & \text{if } x \geq -4 \\ x + 4 & \text{if } x < -4 \end{cases}$$

This piecewise representation accurately reflects the original function but splits it into two linear expressions based on the value of $x$ relative to the critical point $-4$.

Would you like further details or have any other questions on this topic? Here are some possible follow-up questions:

1. How can we graph the function $f(x) = -|x + 4|$?
2. What are the properties of this function such as domain, range, and intercepts?
3. How does the behavior of the function change at $x = -4$?
4. Can we derive similar piecewise functions for other absolute value expressions?
5. How do we verify this piecewise function matches the original function?

Tip: When dealing with absolute value functions, always check for points where the expression inside the absolute value equals zero, as these points often define where the function changes its behavior.