Let \( f(x)=|x-6| \) Use the definition of the absolute value to expand and write the function as a piecewise define function. Choose the most correct answer. (A) \( f(x)=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 0 \\ -x+6 & \text { if } x<0\end{array}\right. \) (B) \( f(x)=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 6 \\ -x+6 & \text { if } x<6\end{array}\right. \) (C) \( f(x)=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 0 \\ -x-6 & \text { if } x<0\end{array}\right. \) (D) \( f(x)=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 6 \\ -x-6 & \text { if } x<6\end{array}\right. \) (E) None of these (D) (D)

To express \( f(x) = |x - 6| \) as a piecewise function, we consider that absolute value reflects the distance from 6: If \( x \geq 6 \), the expression inside the absolute value is non-negative, so \( f(x) = x - 6 \). If \( x < 6 \), the expression is negative, thus \( f(x) = -(x - 6) = -x + 6 \). So the correct function is: \( f(x) = \left\{ \begin{array}{ll} x - 6 & \text{if } x \geq 6 \\ -x + 6 & \text{if } x < 6 \end{array} \right. \) Now for a little fun history: The concept of absolute value has roots in the ancient Greek mathematicians, who were intrigued by the distance of numbers from a certain point unlike the signed number system we commonly use today! As for real-world applications, absolute value functions are critical in various fields like engineering or physics, where we often need to measure distance, regardless of direction. For instance, when calculating the difference in temperature, the actual degree of change matters more than the direction of that change!