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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:
   \[
   f(x) = \begin{cases} 
   - (x + 4) & \text{if } x \geq -4 \\
   - (-x - 4) & \text{if } x < -4 
   \end{cases}
   \]
   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.

Write the following as a piecewise function. f(x) = -|x + 4|

Learn how to write the absolute value function f(x) = -|x + 4| as a piecewise function. This step-by-step explanation covers key concepts like absolute value and how to split a function based on the critical point x = -4. Ideal for students in Grades 9-11.

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