Understanding and plotting $y=|x||y|$

Question

I saw a question on Facebook asking what the following equation looks like:

$$y=|x||y|$$

To be honest, I didn't know where to begin. So I simply graphed it with software and I still couldn't gain any insight. Why is its domain $[-1,1]$?

I tried to generate a table by hand, but with little understanding of the equation, didn't get very far.

Perhaps the question is rather simple, but how does one attack an equation such as this?

Thanks in advance.

Answer 1

The only function $y:\mathbb R\to\mathbb R$ that solves this equation is $y(x)=0$.

Answer 2

The equation does not define a function. In order to plot the graph, break it down by cases:

• if $y = 0$ then the equation is satisfied for all $\forall x\,$, so the entire $x$ axis $y=0$ is part of the graph;

• if $y \gt 0$ then $y = |y|$ and the equation reduces to $|x|=1\,$, so the upper halves $y \gt 0$ of the two vertical lines $x=\pm1$ are also part of the graph;

• if $y \lt 0$ then $y = -|y|$ and the equation reduces to $|x|=-1\,$which has no solutions.