Question

How do you decide whether the relation `x + y^3 = 64` defines a function?

Answer 1

It can be rewritten as `y = root(3)(64-x)`, which uniquely determines `y` for any Real value of `x`, so yes it is a function.

Explanation:

Given `x+y^3=64`, subtract `x` from both sides to get:

`y^3=64-x`

Take cube roots of both sides:

`y = root(3)(64-x)`

This uniquely determines `y` for any Real value of `x`, so the relation does describe a function.

Also note that `x = 64 - y^3` so it has a well defined inverse function too.

Footnote
`x+y^3=64` does not define a function if `x` and `y` are allowed to take Complex values, since every number has `3` cube roots in the Complex plane, hence there would be three possible values of `y` for each value of `x`.