Answer:
[tex]f(x) = \left \{ {{y=\frac{2}{5}(x+5)\ for\ (- \infty, 0) } \atop {y=5-x}\ for\ (4, \infty)} \right.[/tex]
Step-by-step explanation:
[tex]In\ order\ to\ keep\ the\ answer\ simple:\\We'll\ divide\ our\ solution\ into\ 3\ intervals.\\[/tex][tex]In\ the\ interval (- \infty , 0):\\We\ find\ that\ the\ function\ is\ a\ straight\ line\ passing\ through\ the\ points\ (0,2)\\and\ (-5,0).\\Hence, Slope = \frac{y_2 - y_1}{x_2 - x_1} =\frac{0-2}{-5-0} = \frac{2}{5}\\ Using\ the\ point-slope\ formula:\\y-y_0 = m(x-x_0)\\y-0 = \frac{2}{5}(x-(-5))\\ Hence,\\y = \frac{2}{5}(x+5)[/tex]
[tex]In\ the\ Interval\ (0,4):\\The\ graph\ is\ discontinuous\ and\ breaks.[/tex]
[tex]In\ the\ interval (4 , \infty):\\We\ find\ that\ the\ function\ is\ a\ straight\ line\ passing\ through\ the\ points\ (4,1)\\and\ (5,0), although\ (4,1)\ is\ not\ in\ its\ domain.\\Hence, Slope = \frac{y_2 - y_1}{x_2 - x_1} =\frac{0-1}{5-4} = \frac{-1}{1}=-1\\ Using\ the\ point-slope\ formula:\\y-y_0 = m(x-x_0)\\y-0 = -1(x-5)\\ Hence,\\y = 5 - x[/tex]
