To find:
The piecewise function for the graph.
Solution:
From the graph, it is clear that when x is less than -1, the graph passes through (-1, -3) and (-2, -5).
It is known that the equation of a line passes through two points is given by:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
So, the equation of line passing through (-1, -3) and (-2, -5) is:
[tex]\begin{gathered} y-(-3)=\frac{-5-(-3)}{-2-(-1)}(x-(-1)) \\ y+3=\frac{-2}{-1}(x+1) \\ y+3=2x+2 \\ y=2x-1 \end{gathered}[/tex]
So, the first drop down is "2x - 1", and second drop down is "x is less than or equal to -1".
Now, the graph passes through (1, 5) and (2, 6). So, the equation of the line is:
[tex]\begin{gathered} y-5=\frac{6-5}{2-1}(x-1) \\ y-5=x-1 \\ y=x+4 \end{gathered}[/tex]
So, the third drop down menu is "x + 4" and the fourth drop down menu is "x is greater than or equal to 1".
