There will be either a hole or a vertical asymptote at the x-value that makes the denominator zero.
Holes: The graph does not have any holes. Vertical Asymptotes: x=-2/3 and x=-1
Practice makes perfect
Consider the given function.
y=x-1/(3x+2)(x+1)
Recall that division by zero is not defined. Therefore, the rational function is undefined where (3x+2)(x+1)=0.
ccc
3x+2=0 & & x+1=0
⇕ & & ⇕
x=-2/3 & & x=-1
The function is not defined when x=- 23 and x=-1. This means we have either a hole or a vertical asymptote. Note that, the function is already factored and simplified.
Therefore, all the factors of the denominator are still there. This indicates the lines x=- 23 and x=-1 are vertical asymptotes.
