A rational function can have three types of asymptotes: horizontal, vertical, and slant asymptotes. Apart from these, it can have holes as well. Let us see how to find each of them.
Holes of a Rational Function
The holes of a rational function are points that seem that they are present on the graph of the rational function but they are actually not present. They can be obtained by setting the linear factors that are common factors of both numerator and denominator of the function equal to zero and solving for x. We can find the corresponding y-coordinates of the points by substituting the x-values in the simplified function. Every rational function does NOT need to have holes. Holes exist only when numerator and denominator have linear common factors.
Example: Find the holes of the function f(x) = (x2 + 5x + 6) / (x2 + x – 2).
Solution:
Let us factorize the numerator and denominator and see whether there are any common factors.
f(x) = [ (x + 2)(x + 3) ] / [ (x + 2) (x – 1) ]
= [ ̶(̶x̶ ̶+̶ ̶2̶)̶(x + 3) ] / [ ̶(̶x̶ ̶+̶ ̶2̶)̶ (x – 1) ]
= (x + 3) / (x – 1)
Since (x + 2) was striked off, there is a hole at x = -2. Its y-coordinate is f(-2) = (-2 + 3) / (-2 – 1) = -1/3.
Thus, there is a hole at (-2, -1/3).