Answer:
The piecewise function is
[tex]f(x)=\begin{cases}2x-1&\text{ if }x\leq-1\\ x+4&\text{ if } x\geq1 \end{cases}[/tex]
Step-by-step explanation:
From the given graph it is noticed that the graph is divides into two pieces.
The first function is defined for all values of x which are less than or equal to -1. The second function is defined for all values of x which are greater than or equal to 1.
The equation of line which passing through two points is
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
The first line is passing trough (-1,-3) and (-2,-5).
[tex]y-(-3)=\frac{-5-(-3)}{-2-(-1)}(x-(-1))[/tex]
[tex]y+3=\frac{-2}{-1}(x+1)[/tex]
[tex]y+3=2(x+1)[/tex]
[tex]y=2x+2-3[/tex]
[tex]y=2x-1[/tex]
For [tex]x\leq-1[/tex] the function is defined as,
[tex]f(x)=2x-1[/tex]
Similarly the second line is passing through (1,5) and (2,6).
[tex]y-5=\frac{6-5}{2-1}(x-1)[/tex]
[tex]y-5=1(x-1)[/tex]
[tex]y=x+4[/tex]
For [tex]x\geq-1[/tex] the function is defined as,
[tex]f(x)=x+4[/tex]
Therefore, the piecewise function is,
[tex]f(x)=\begin{cases}2x-1&\text{ if }x\leq-1\\ x+4&\text{ if } x\geq1 \end{cases}[/tex]
