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I am graphing a square with the following equation:

$$|y|=1-|x|$$

enter image description here

However, I need the equation in terms of y. That is, the form y=f(x) as opposed to the current |y|=f(x)

How do you get an equation in terms of y when absolute value is wrapping it. Is there an inverse absolute value function I can apply to both sides of the equation?


My first attempt was to use the old trick: $|n| = \sqrt[2]{n^2}$

But of course that's not a real equation since roots have $±$.

Sure enough when applying that to try to get the equation in terms of y the graph failed to reproduce a square and ended up graphing what looked like a 'w'.

enter image description here

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    $\begingroup$ This isn't a function, so the only way to do it would be piecewise-ish. $\endgroup$ – Randall Dec 17 '18 at 18:50
  • $\begingroup$ @Randall I wonder if there is a way to do it without going piecewise. I am currently experimenting with converting absolute value to this form $|y| \rightarrow y\cdot sgn(y)$ $\endgroup$ – Albert Renshaw Dec 17 '18 at 18:54
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    $\begingroup$ But sgn(y) is also piecewise... run into same problem. $\endgroup$ – coffeemath Dec 17 '18 at 18:58
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    $\begingroup$ @AlbertRenshaw It is not possible. A single $x$ creates two different $y$s (generally), so there is no way to make this function-like. $\endgroup$ – Randall Dec 17 '18 at 19:01
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    $\begingroup$ Then the answer is no, since you have two branches. That's similar to the equation of a circle $x^2+y^2=1$ $\endgroup$ – Andrei Dec 17 '18 at 19:08
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$y = \pm |y| = \pm (1 - |x|)$ for $-1 \le x \le 1$.

That is, there are two cases: $+(1 - |x|)$ which gives you the top half of your square, and $-(1-|x|)$ which gives you the bottom half.

Those inequalities are necessary, because $|y|$ is not allowed to be negative.

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  • $\begingroup$ Thanks, I was hoping for something unbounded but this will have to do for now. +1 $\endgroup$ – Albert Renshaw Dec 17 '18 at 19:23
  • $\begingroup$ You cannot do any better, and this still is not of the form $y=f(x)$ that you originally asked for, because that's impossible. The answer here is the best possible, but please do not think it is a function. $\endgroup$ – Randall Dec 17 '18 at 20:43
  • $\begingroup$ @randall you’re getting too caught up on the word “function”. I’m aware function isn’t the correct term, however this answer is sufficient for the use-case I have. I’m leaving the question open for now to see what else others may come up with $\endgroup$ – Albert Renshaw Dec 17 '18 at 21:26
  • $\begingroup$ What do you mean "something unbounded"? The square is bounded! $\endgroup$ – Robert Israel Dec 17 '18 at 21:55

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