3:29C Previous Next Unit Pretest 5 Select the correct answer. Consider this absolute value function. \[ f(x)=|x-5| \] How can function \( f \) be written as a piecewise function? A. \( f(x)=\left\{\begin{array}{ll}x-5, & x \geq 5 \\ -x-5, & x<5\end{array}\right. \) B. \( f(x)=\left\{\begin{array}{ll}x-5, & x \geq-5 \\ -x-5, & x<-5\end{array}\right. \) C. \( f(x)=\left\{\begin{array}{ll}x-5, & x \geq 5 \\ -x+5, & x<5\end{array}\right. \) D. \( f(x)=\left\{\begin{array}{ll}x-5, & x \geq-5 \\ -x+5, & x<-5\end{array}\right. \)

To express the absolute value function \( f(x) = |x - 5| \) as a piecewise function, we need to consider where the expression inside the absolute value changes sign. For values of \( x \) that are greater than or equal to 5, \( f(x) \) simplifies to \( x - 5 \). For values of \( x \) less than 5, \( f(x) \) simplifies to \( -(x - 5) \) or equivalently \( -x + 5 \). Thus, the correct answer is: C. \( f(x)=\left\{\begin{array}{ll}x-5, & x \geq 5 \\ -x+5, & x<5\end{array}\right. \) This means that for \( x \) greater than or equal to 5, we just subtract 5, and for \( x \) less than 5, we reflect it upwards by adding 5!