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I want to calculate the equipotential lines for $f(x, y) = x + y + |x| + |y|$. The domain is $ℝ^2$ and range $[0, \infty)$. I started like this:

$$ x + y + |x| + |y| = c \ge 0 \\ y + |y| = c - x - |x| $$

But I can't continue any further, I don't know how to get rid of the $|y|$. How can I solve the last equation by $y$?

Michael Hardy
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Chris
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  • We have that for non-negative values $|v| + v = 2v$ and for negative values of $v$, we have that $|v| + v = 0$. Split into cases. – Eff Jul 04 '16 at 23:24
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    Solve separately the four cases $$x\ge0\wedge y\ge0\x<0\wedge y\ge 0\x\ge0\wedge y<0\ x<0\wedge y<0$$ and see the simplifications. –  Jul 04 '16 at 23:25
  • Consider the function $f(y)=y+|y|$. On positive real numbers it simplifies and is obviously injective, but on non-positive real numbers it is not injective. Indeed it simplifies there to a constant. – hardmath Jul 04 '16 at 23:26

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