Let \( f(x)=|x|-3 \) Use the definition of the absolute value to expand and write the function as a piecewise define function. Choose the most correct answer. (A) \( f(x)=\left\{\begin{array}{ll}x-3 & \text { if } x \geq 0 \\ -x+3 & \text { if } x<0\end{array}\right. \) (B) \( f(x)=\left\{\begin{array}{ll}x-3 & \text { if } x \geq 3 \\ -x+3 & \text { if } x<3\end{array}\right. \) (C) \( f(x)=\left\{\begin{array}{ll}x-3 & \text { if } x \geq 0 \\ -x-3 & \text { if } x<0\end{array}\right. \) (D) \( f(x)=\left\{\begin{array}{ll}x-3 & \text { if } x \geq 3 \\ -x-3 & \text { if } x<3\end{array}\right. \) (E) None of these

To transform \( f(x) = |x| - 3 \) into a piecewise function, we start by applying the definition of absolute value. For \( x \geq 0 \), \( |x| = x \), so: - If \( x \geq 0 \): \( f(x) = x - 3 \) For \( x < 0 \), \( |x| = -x \), giving us: - If \( x < 0 \): \( f(x) = -x - 3 \) Thus, the correct piecewise function representation is: \[ f(x) = \begin{cases} x - 3 & \text{if } x \geq 0 \\ -x - 3 & \text{if } x < 0 \end{cases} \] None of the provided options perfectly matches this. Therefore, the answer is (E) None of these. Now, for some fun insights: The concept of absolute value dates back to around the time of the ancient Greeks, who explored the idea of distance and quantity. The absolute value function helps us measure how far a number is from zero, regardless of direction—think of it as finding the shortest path home without worrying about the direction taken! Understanding piecewise functions is crucial in real-world scenarios, such as in economics when modeling costs. For instance, a shipping company may charge different rates based on weight thresholds, necessitating a piecewise function to accurately represent pricing depending on the weight brackets!