Question

How do you decide whether the relation `x^2-3y =12` defines a function?

Answer 1

A function is when for every `x` there is a maximum of only one `y`

Explanation:

Let's rewrite the relation by adding `3y` to both sides and subtracting `12` (also on both sides) and then divide by `3`:

`x^2-3y=12->x^2-12=3y->y=1/3x^2-4`
graph{x^2/3-12 [-28.86, 28.87, -14.43, 14.43]}
This gives one `y`-value for every `x`, so this may be called `y(x)`, or `y` as a function of `x`.

Note :
You can not write `x(y)`, or `x` as a function of `y`, because for many `y`'s there are two possible `x`'s.