Does the equation X equals the absolute value of Y define a function? And why?

Answer 1

No

First I will answer in the language of math and then I will explain in English.

It is not a function because for each element in the domain there is more than one element in the range.

What that means is that for every X there is more than one Y. In this case there are 2 Ys for every X..

Rearrange to the absolute value of Y equals X.

That means Y equals plus or minus any X.

For example, for X = 5, both Y = 5 and Y = -5 satisfy the equation.

If you were to graph this you would have a sideways V shape that would obviously not pass the vertical line test.