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Domain of Rational Functions
A rational function is a function that looks like a fraction and has a variable in the denominator. The following are examples of rational functions:


Note that a function such as is not considered a rational function. Even though it is in the form of a fraction, the denominator does not contain a variable.

Whenever we are dealing with fractions, we are not allowed to have zero in the denominator. Anytime zero is in the denominator of a fraction, we have something that is undefined. So when dealing with rational functions, we have to make sure the denominator is never equal to zero.

The domain of a function consists of the numbers we are allowed to use for the variable in that function. So with rational functions, if there is a number that will cause the denominator of the function to be equal to zero, we need to exclude it from our domain.

Examples
Example
#1:
  • We do not want the denominator of this function to ever equal zero. The only time this would happen is when or .
In general, if we set the denominator of the function equal to zero and solve the equation, we can find out what values cannot be used in the domain of the function.
What is your answer?
 
Example
#2:
What is your answer?
 
Example
#3:
  • Let . Solving this equation requires factoring. If you need to learn more about factoring, click here to go to the factoring lesson.

  • To solve this equation, we need to set each factor equal to zero. If you need to see more on solving quadratics, click here to go to the solving quadratics lesson.
What is your answer?
 
Example
#4:
  • When we set and solve the equation, we get . In other words, we have two imaginary solutions to the equation.

  • Keep in mind that when trying to find the domain of these functions, we only considered the denominator. When you begin graphing rational functions, you will work with both the numerator and denominator to determine the graph of the function. However, it is important to realize that determining the domain only requires excluding values that make the denominator zero. 
What is your answer?
 



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