Question

Does the equation `abs(y-x) = 1` define a function?

Answer 1

The relation `abs(y-x)=1` does not describe a function since it does not result in a unique value of `y` for each value of `x` in its domain.

Explanation:

A relation is a function if it defines at most one value of `y` for each value of `x`.

The equation:

`abs(y-x) = 1`

describes a relation between `x` and `y` which does not result in a unique value of `y` for all values of `x`. In fact for any `x` there are two possible values of `y`.

Here is its graph:

graph{abs(y-x)=1 [-5, 5, -2.5, 2.5]}

So for example, if `x=0` then `y = 1` or `y = -1`