2.7. Holes in Rational Functions http://www.ck12.org
Note that there are two holes that do not fit in the graph window. When this happens you still need to note where
they would appear given a properly sized window. To do this, substitute the invalidxvalues:x=− 1 ,− 2 ,3 into the
factored cubic that remained after canceling.
f(x) = (x− 2 )^3 − 1
Holes: (3, 0); (-1, -28); (-2, -65)
Concept Problem Revisited
f(x) =(^3 x+(x^1 −)( 1 x)−^1 )
For this function that is not defined atx=1 there is a removable discontinuity that is represented as a hollow circle
on the graph. Otherwise the function behaves precisely as 3x+1.
Vocabulary
Aremovable discontinuity,also known as a hole, is a point on a function that occurs because a factor can be canceled
from the numerator and the denominator of the rational function.
Arational functionis a function with at least one rational expression.
Arational expressionis a ratio of two polynomial expressions.
Guided Practice
- Without graphing, identify the location of the holes of the following function.
f(x) =x^3 +x (^24) +x 52 +x+x− 66
- What is a possible equation for the following rational function?