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I am asked to create a piecewise representation of the function below but I don't really understand how I would get multiple equations and conditions from that single function? All I know is that the domain of the entire given function would be (-∞, ∞) but I don't think that's the correct answer to the question.

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You need two pieces to the puzzle. One is this: $$|x|=\begin{cases}&x,\;x\ge 0 \\ -\!\!\!\!\!&x, \;x<0\end{cases}.$$ The other is solving $x^2+7x+6=0.$ Can you put these two together and finish?

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