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

May 29, 2018

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: $3 y - 1 = 7 x + 2$, the dependent variable is $y$ and the independent variable is $x$.

The given relation can be re-arranged as
$\textcolor{w h i t e}{\text{XXX}} y = \frac{7 x + 3}{3}$
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 $x$.