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: