\( -4|6-3 x|+8 \) write as a Picewuse define function

To express the function \( f(x) = -4|6-3x| + 8 \) as a piecewise function, you need to consider two cases based on the absolute value expression. The critical point is when the inside of the absolute value is equal to zero, which occurs at \( 6 - 3x = 0 \) or \( x = 2 \). 1. If \( x < 2 \), then \( 6 - 3x > 0 \) and \( |6 - 3x| = 6 - 3x \). Substituting this into the function gives: \[ f(x) = -4(6 - 3x) + 8 = -24 + 12x + 8 = 12x - 16 \] 2. If \( x \geq 2 \), then \( 6 - 3x \leq 0 \) and \( |6 - 3x| = -(6 - 3x) = 3x - 6 \). Thus: \[ f(x) = -4(3x - 6) + 8 = -12x + 24 + 8 = -12x + 32 \] Putting it all together, you have: \[ f(x) = \begin{cases} 12x - 16 & \text{if } x < 2 \\ -12x + 32 & \text{if } x \geq 2 \end{cases} \]