Module 7: Rational Functions and Function Operations

 

The main difference between the two functions in Try This 2 is that the non-permissible value for one produced a point of discontinuity and the other produced an asymptote.

 

  When a factor that occurs in both the numerator and denominator is equal to 0 When a factor that occurs only in the denominator is equal to 0
Function
Non-permissible Values

x = −1

 

x ≠ −1

 
Type of Non-permissible Value indeterminate undefined
Appearance on Graph

hole or point of discontinuity

 

This is a diagram of a function with a hole.

asymptote

 

This is a diagram of a function with an asymptote.
Factored Form

Common Factors

 

 Yes, a factor in the numerator and in the denominator causes a hole. No, a factor only in the denominator causes an asymptote.

 

This diagram shows a point of discontinuity and states “Points of discontinuity or ‘holes’ occur when a factor in both the numerator and denominator equals 0.” It also shows an asymptote and states “Asymptotes occur when a factor only in the denominator equals 0.”

 

Did You Know?

A formal definition of division is  if and only if a = bc. A division by 0 fails this definition. If you begin with , then a = 0c also needs to be true—it is not for any a ≠ 0. This is why a division by 0 is usually said to be undefined.

 

An exception occurs when a = 0. This gives  , or 0 = 0c, which is true for any c. By this definition,  is equal to any value.  is often referred to as indeterminate.



You do not need to understand indeterminate values for this course. You only need to recognize that indeterminate values will cause points of discontinuity to appear on the graph of a rational function.