Chapter 5.5, Problem 78STP

To determine

To find: the piecewise function from the given graph.

Answer to Problem 78STP

  $\text{f}\left(\text{x}\right)\text{=}\left\{\begin{array}{cc}-4& \text{x}\le \text{-2}\\ -\text{2x+3}& -\text{2<x<1}\\ \text{x-5}& \text{x}\ge \text{2}\end{array}\right\}$ .

Explanation of Solution

Given:

  

Concept used:

According to the Straights line equation for one point:

  ${\text{y-y}}_1\text{=m}\left({\text{x-x}}_1\right)$ .

  $\text{slope}\left(\text{m}\right)\text{=}\frac{\text{dy}}{\text{dx}}\text{=}\frac{{\text{y}}_{\text{2}}{\text{-y}}_{\text{1}}}{{\text{x}}_{\text{2}}{\text{-x}}_{\text{1}}}\text{=tanθ}$ .

Based on the value of domain $\text{x}$ function will behave differently.

Or

Multiple sub function applies to a different sub domain, a certain interval of the main function’s domain and are grouped together.

Calculation:

From the given graph:

For $\text{x}\le \text{2}$:

According to the Straights line equation formula:

  $\text{y=mx+c}$

  $\text{y=-4}$ .

For $\text{-2<x<1}$:

Slope:

  $\text{m=-2}$ .

According to the Straights line equation for one point:

  ${\text{y-y}}_1\text{=m}\left({\text{x-x}}_1\right)$ .

  $\text{y-3=-2}\left(\text{x-0}\right)$ .

For $\text{x}\ge \text{2}$ :

Here the slope $\text{m=1}$ .

According to the Straights line equation for one point:

  ${\text{y-y}}_1\text{=m}\left({\text{x-x}}_1\right)$ .

  $\text{y-0=1}\left(\text{x-5}\right)$ .

Hence, the function for the given graph is:

  $\text{f}\left(\text{x}\right)\text{=}\left\{\begin{array}{cc}-4& \text{x}\le \text{-2}\\ -\text{2x+3}& -\text{2<x<1}\\ \text{x-5}& \text{x}\ge \text{2}\end{array}\right\}$ .