Chapter 4.7, Problem 43E

To determine

Write the absolute value function as piecewise function.

Answer to Problem 43E

  $\begin{array}{l}y=5x-40\text{if}x<8\text{and}\\ y=-5x+40\text{if}x\ge 8.\end{array}$

Explanation of Solution

Given information:Given absolute function is $y=-5\left|x-8\right|.$

Calculation:

Formula:

Absolute value function $f(x)=a\left|x-h\right|+k$ can be written as piecewise function as,

  $\begin{array}{l}f(x)=a\left[-(x-h)\right]+k\text{if}x-h<0\text{and}\\ f(x)=a(x-h)+k\text{if}x-h\ge 0.\end{array}$

Here in given absolute function $a=-5\text{and}h=8\text{and}k=0.$ So piecewise function is,

  $\begin{array}{l}y=-5\left[-(x-8)\right]+0\text{if}x-8<0\text{and}\\ y=-5\left[x-8)\right]+0\text{if}x-8\ge 0.\\ y=-5\left[-x+8)\right]\text{if}x<8\text{and}\\ y=-5\left[x-8\right]\text{if}x\ge 8.\\ y=5x-40\text{if}x<8\text{and}\\ y=-5x+40\text{if}x\ge 8.\end{array}$

So from absolute function $y=-5\left|x-8\right|.$

Piecewise function is ,

  $\begin{array}{l}y=5x-40\text{if}x<8\text{and}\\ y=-5x+40\text{if}x\ge 8.\end{array}$