Write a piecewise function for the graph.

The line to the left of the breakpoint (x=-1) has a rise of 2 for each run of 1, so its slope is rise/run = 2/1 = 2. We can use the point-slope formula to write its equation, considering it goes through the point (-2, -2).

y -(-2) = 2(x -(-2))

y = 2x +4 -2 . . . . . eliminate parentheses, subtract 2

y = 2x +2 . . . . . . . equation for x ≤ -2

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The line to the right of the breakpoint has a rise of 1 for each run of 2. Its slope is rise/run = 1/2. For this portion of the graph, we can read the y-intercept from the graph: -1. The equation for the function for x > -2 is ...

y = 1/2x -1 . . . . . . . equation for x > -2

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We note that both of these lines could include the point (-2, -2), but that is specifically excluded from the right-side equation by its condition x > -2. So, we must include that point in the left-side equation by making the condition include the case x=-2.

[tex]f(x)=\left\{\begin{array}{ll}2x+2&\text{, if $x\le-2$}\\\dfrac{1}{2}x-1&\text{, if $x>-2$}\end{array}\right.[/tex]