Question

Write a piecewise function for the graph. ​

Answer 1

so let's use a couple of points on the line for the two slanted lines to get their equations, say for the first one with the arrowhead, we'll use the points (-1 , 3) and (-3 , 1)

`(\stackrel{x_1}{-1}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{-3}~,~\stackrel{y_2}{1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{1}-\stackrel{y1}{3}}}{\underset{\textit{\large run}} {\underset{x_2}{-3}-\underset{x_1}{(-1)}}} \implies \cfrac{ -2 }{-3 +1} \implies \cfrac{ -2 }{ -2 } \implies 1`

`\begin{array}c \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{3}=\stackrel{m}{1}(x-\stackrel{x_1}{(-1)}) \implies y -3 = 1 ( x +1) \\\\\\ y -3 = x +1 \implies \boxed{y = x +4}`

now for the other one, we'll use (-1 , 0) and (3 , -1)

`(\stackrel{x_1}{-1}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-1}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{(-1)}}} \implies \cfrac{ -1 }{3 +1} \implies \cfrac{ -1 }{ 4 } \implies -\cfrac{1}{4}`

`\begin{array} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{0}=\stackrel{m}{-\cfrac{1}{4}}(x-\stackrel{x_1}{(-1)}) \\\\\\ y = -\cfrac{1}{4} ( x +1) \implies \boxed{y=-\cfrac{1}{4}x-\cfrac{1}{4}}`

and the last line, the horizontal one, well, that's just y = -3.

the 1st line goes to -infinity and up to -1

the 2nd line goes from -1 up to 3 but exempting the endpoints

the 3rd one takes it from there to +infinity.

`f(x)= \begin{cases} x+4&x~ \leqslant ~-1\\\\ -\cfrac{1}{4}x-\cfrac{1}{4} ~~ ~~ &-1 < x < 3\\\\ -3&x~ \geqslant ~ 3 \end{cases}`