Question

How do you decide whether the relation `3y - 1 = 7x +2` defines a function?

Answer 1

You need to consider the question: Does this relation allow for multiple values of the dependent variable for any value of the independent variable?

Explanation:

For demonstration purposes I will assume that in the given relation: `3y-1=7x+2`, the dependent variable is `y` and the independent variable is `x`.

The given relation can be re-arranged as
`color(white)("XXX")y=(7x+3)/3`
and we can see (hopefully this is obvious) that any value of `x` will provide one and only one value for `y`.

Since this was asked under the topic "Functions on a Cartesian Plane", another way to look at this is (assuming you have some way of generating the graph of this relation) is to see if there is any possibility of a vertical line crossing the line of this equation in more than one place:

In this case we can see that the graph is a straight line, so there can not be any "doubling back" to provide more than one value of `y` for any value of `x`.