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

1 Answer
Jun 26, 2015

Answer:

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]}