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The given function is \( f(x) = |4x + 1| \), and the question asks you to rewrite this absolute value function as a piecewise-defined function. 

To write this as a piecewise function, we need to consider the two cases for the expression inside the absolute value, \( 4x + 1 \), based on when it is positive or negative.

### Step-by-Step Process:
1. **Find when \( 4x + 1 = 0 \):**
   Solve \( 4x + 1 = 0 \) to find the point where the expression inside the absolute value changes sign.
   \[
   4x = -1 \quad \Rightarrow \quad x = -\frac{1}{4}
   \]
   So, the expression \( 4x + 1 \) is non-negative when \( x \geq -\frac{1}{4} \) and negative when \( x < -\frac{1}{4} \).

2. **Define the piecewise function:**
   - When \( x \geq -\frac{1}{4} \), \( 4x + 1 \) is non-negative, so \( |4x + 1| = 4x + 1 \).
   - When \( x < -\frac{1}{4} \), \( 4x + 1 \) is negative, so \( |4x + 1| = -(4x + 1) = -4x - 1 \).

Thus, the piecewise-defined function is:
\[
f(x) = \begin{cases} 
      -(4x + 1), & \text{if } x < -\frac{1}{4} \\
      4x + 1, & \text{if } x \geq -\frac{1}{4}
   \end{cases}
\]

This matches the form you're asked to fill in, where you split the absolute value function based on the critical point \( x = -\frac{1}{4} \).

Would you like more details on piecewise functions or the absolute value process?

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Here are five related questions you might find useful:
1. How do you graph piecewise-defined functions?
2. What are the key points to consider when rewriting absolute value functions in piecewise form?
3. How can inequalities be used to define different regions for a function?
4. What happens if the expression inside the absolute value has more than one root?
5. How do you determine the domain and range of piecewise-defined functions?

**Tip**: Always check for the point where the expression inside the absolute value equals zero, as this is where the function's definition will change.

Write the absolute value function as a piecewise-defined function with linear parts: f(x) = |4x + 1|

Learn how to write the absolute value function f(x) = |4x + 1| as a piecewise-defined function with linear parts. This solution covers key concepts in absolute value and piecewise functions, suitable for students in Grades 9-12.

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