Question

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

Answer 1

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 - xy + y + 1 = x + 1 - (x - 1)y`

Add `(x-1)y` to both ends to get:

`(x-1)y = (x+1)`

Divide both sides by `(x-1)` to get:

`y = (x+1)/(x-1)`

For any `x != 1` this determines the value of `y` uniquely. So a vertical line of the form `x=a` with `a != 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]}