Use the definition of the absolute value to expand and write the function \( f(x)=|x-4|-5 \) as a piecewise define function. Choose the most correct answer. (A) \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 0 \\ -x-1 & \text { if } x<0\end{array}\right. \) (B) \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 4 \\ -x-1 & \text { if } x<4\end{array}\right. \) (B) \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 0 \\ -x-9 & \text { if } x<0\end{array}\right. \) (C) \( f \) REMAINING (D) \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 4 \\ -x-9 & \text { if } x<4\end{array}\right. \) (E) None of these

To express the function \( f(x) = |x - 4| - 5 \) as a piecewise defined function, we need to first deal with the absolute value term \( |x - 4| \). The definition of absolute value leads us to two cases based on the value of \( x \): 1. When \( x - 4 \geq 0 \) (or \( x \geq 4 \)), \( |x - 4| = x - 4 \). 2. When \( x - 4 < 0 \) (or \( x < 4 \)), \( |x - 4| = -(x - 4) = -x + 4 \). Now substituting these into the function gives us: - For \( x \geq 4 \): \[ f(x) = (x - 4) - 5 = x - 9 \] - For \( x < 4 \): \[ f(x) = (-x + 4) - 5 = -x - 1 \] So, we can write the piecewise function as: \[ f(x) = \begin{cases} x - 9 & \text{if } x \geq 4 \\ -x - 1 & \text{if } x < 4 \end{cases} \] The most correct answer is **(D)** \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 4 \\ -x-1 & \text { if } x<4\end{array}\right. \). Feeling intrigued? The concept of absolute values stems from the need to deal with numbers in a directional way, ensuring we can always talk about distances or magnitudes, irrespective of whether the original number was positive or negative. This fundamental notion is pivotal in mathematics where understanding the concept of "size" without sign opens doors to various applications! Speaking of applications, the piecewise function representation is useful in many real-world scenarios! For instance, it can model situations where fixed costs change based on certain thresholds, such as a telecommunications plan where charges vary depending on usage. Understanding how to break down functions like this can help you analyze situations involving varying conditions, ultimately leading to better decisions!