\( |x-5|+7 \) as Picewise

The expression \( |x-5| + 7 \) can be written as a piecewise function by considering the definition of absolute value. For \( x \geq 5 \), the absolute value \( |x-5| \) simplifies to \( x-5 \). Thus, the function becomes \( (x-5) + 7 = x + 2 \). For \( x < 5 \), the absolute value \( |x-5| \) becomes \( -(x-5) \), leading to \(-x + 5 + 7 = -x + 12\). So, we have: \[ f(x) = \begin{cases} x + 2 & \text{if } x \geq 5 \\ -x + 12 & \text{if } x < 5 \end{cases} \] This encapsulates how the function behaves across different intervals of \( x \).