Use the definition of the absolute value to expand and write the function \( f(x)=|x-4|-5 \) as a plecewise 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) \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 4 \\ -x-1 & \text { if } x<4\end{array}\right. \) (C) \( f(x)=\left\{\begin{array}{ll}x-9 & \text { if } x \geq 0 \\ -x-9 & \text { if } x<0\end{array}\right. \) (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 first recognize that the absolute value function \( |x-4| \) has different expressions based on the value of \( x \). Specifically, if \( x \geq 4 \), then \( |x-4| = x-4 \), and if \( x < 4 \), then \( |x-4| = -(x-4) = -x + 4 \). Now, we can use these conditions to rewrite \( f(x) \): 1. For \( x \geq 4 \): \( f(x) = (x-4) - 5 = x - 9 \) 2. For \( x < 4 \): \( f(x) = (-x + 4) - 5 = -x - 1 \) Therefore, the piecewise function is: \[ f(x) = \begin{cases} x - 9 & \text{if } x \geq 4 \\ -x - 1 & \text{if } x < 4 \end{cases} \] The correct answer is: (B) \( f(x) = \left\{\begin{array}{ll}x-9 & \text { if } x \geq 4 \\ -x-1 & \text { if } x<4\end{array}\right. \) Now, let's expand on this topic! The concept of absolute value has a rich historical background. It originates from the need to measure distances, ensuring that outcomes are always non-negative, paving the way for establishing fundamental algebraic principles. Ancient mathematicians recognized that thinking of distances in negative terms didn’t make sense, leading to the definition of absolute value we use today! In the real world, the application of absolute value is everywhere! For instance, it's critical in fields like engineering and finance, where the difference in quantities (like profit and loss) needs to be assessed without the chaos of negative values interfering. It helps you understand how far you are from a baseline, such as zero debt or loss, regardless of direction!