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=- 12 and x=4
Practice makes perfect
Consider the given function.
y = x+2/(2x+1)(x-4)
Recall that division by zero is not defined. Therefore, the rational function is undefined where 2x+1=0 or x-4 = 0.
ccc
2x+1=0 & & x-4=0
⇕ & & ⇕
x=-1/2 & & x=4
The function is not defined when x=- 12 or x = 4. This means we have either holes or vertical asymptotes. Note that the function is simplified as much as possible. We have the factors 2x+1 and x-4 in the denominator. This indicates that the lines x = - 12 and x = 4 are vertical asymptotes.
