# How do you use the vertical line test to show x-xy+y+1=0 is a function?

Jun 26, 2015

Show that there is a unique $y$ value for each $x$ value in the domain, so any vertical line will only cut the curve at one point.

#### Explanation:

$0 = x - x y + y + 1 = x + 1 - \left(x - 1\right) y$

Add $\left(x - 1\right) y$ to both ends to get:

$\left(x - 1\right) y = \left(x + 1\right)$

Divide both sides by $\left(x - 1\right)$ to get:

$y = \frac{x + 1}{x - 1}$

For any $x \ne 1$ this determines the value of $y$ uniquely. So a vertical line of the form $x = a$ with $a \ne 1$ will cut the curve at exactly one point.

How about $x = 1$?

Then the original equation becomes:

$0 = 1 - y + y + 1 = 1 + 1 = 2$

which is false, so there are no points on the curve with $x = 1$.

graph{x-xy+y+1 = 0 [-10, 10, -5, 5]}