1. Chapter 12
Rational
Functions

2. Section 12.1
Finding the Domains of Rational Functions and Simplifying Rational Expressions

3. Meaning of a Rational Function
Definition of a Rational Function
Definition
A rational function is a function whose equation can be put into the form
where P(x) and Q(x) are polynomials and Q(x) is nonzero.

4. Evaluate the function at the indicated values. 1. f (6) 2. f (–3)
Meaning of a Rational Function
Definition of a Rational Function
Example
Evaluate the function at the indicated values.
1. f (6) 2. f (–3)
Solution

5. Domain of a Rational Function
Definition of a Rational Function
Example
Find the domain of the function.
1. 3 is not in the domain because
Solution
Domain: { x | x ≠ 3 }

6. Domain of a Rational Function
Definition of a Rational Function
Solution Continued
Example 2.
Domain: { x | x ≠ -2, 2 }

7. Domain of a Rational Function
Property
The domain of a rational function
is the set of real numbers except for those numbers that, when substituted for x, give Q(x) = 0.
Find the domain of the function.
Example

8. Domain of a Rational Function
Solution
1. Set the denominator equal to 0 and solve
Set the denominator equal to 0 and solve
Domain: All real numbers
Domain: { x | x ≠ }

9. Domain of a Rational Function
Excluded Value
Definition
A number is an excluded value of a rational expression if substituting the number into the expression leads to a division by 0.

10. Simplifying Rational Expressions
Example of Simplifying Rational Expressions
Example
Simplifying

11. 3 and –3 are excluded values
Simplifying Rational Expressions
Example of Simplifying Rational Expressions
Example Continued
Example
3 and –3 are excluded values

12. Simplifying Rational Expressions
Example of Simplifying Rational Expressions
Property
To simplify a rational expression,
1. Factor the numerator and the denominator.
Use the property
where A and C are nonzero, so that the expression is in lowest terms.

13. Simplify the right-hand side
Simplifying Rational Expressions
Example of Simplifying Rational Expressions
Example
Simplify the right-hand side
Solution

14. Simplifying Rational Expressions
Example of Simplifying Rational Expressions
Solution Continued
Example
Domain is all real numbers except –5 and 5
Check by substituting and graphing calculator
Domain: { x | x ≠ -5, 5 }

15. Property Example Connection Between Vertical Asymptotes and Domains
Vertical Asymptotes and Domains of Rational Functions
Property
If a rational function g has a vertical asymptote x = k, then k is not in the domain of g. If k is not in the domain of a rational function h, then x = k may or may not be a vertical asymptote of h.
Simplify the right-hand side of the equation
Example

16. Solution Connection Between Vertical Asymptotes and Domains
Simplifying Right-Hand Side of an Equation
Solution

17. Simplify the right-hand side of the equation
Connection Between Vertical Asymptotes and Domains
Simplifying Right-Hand Side of an Equation
Example
Simplify the right-hand side of the equation
Solution

18. Solution Continued Example
Connection Between Vertical Asymptotes and Domains
Simplifying Right-Hand Side of an Equation
Solution Continued
Example

19. Example Solution Connection Between Vertical Asymptotes and Domains
Simplifying Right-Hand Side of an Equation
Example
Simplify
Solution

20. Definition of the Quotient Function
If f and g are functions, x is in the domain of both functions, and g(x) is nonzero, then we can form the
quotient function
Let
1. Find an equation of Simplify the right-hand side of the equation.
2. Find
Example

21. Quotient Function
Quotient Functions
Solution

22. Use a Rational Model to Describe the Percentage of Quantity
Percentage Formula
Formula
If m items out of n items have a certain attribute, then the percentage p (written p%) of the n items that have the attribute is
We call this equation the percentage formula.
In Exercise 5 of Homework 2.1, you modeled the number of Internet users in the United States (see the table on the next slide).
Example

23. Example Continued Formula
Use a Rational Model to Describe the Percentage of Quantity
Using a Rational Model to Make a Prediction
Example Continued
Formula
A reasonable model is I (t)=20.72t −83.94, where I (t) is the number of Internet users (in millions) at t years since In Exercise of Section 7.7, you modeled the U.S population. A reasonable model is U(t) = t t , where U(t) is the U.S. population (in millions) at t years since 1990.

24. Example Continued Formula
Use a Rational Model to Describe the Percentage of Quantity
Using a Rational Model to Make a Prediction
Example Continued
Formula
1. Let P(t) be the percentage of Americans who are Internet users at t years since Find an equation of P.
2. Use P to estimate the percentage of Americans who were Internet users in Then compute the actual percentage by referring to the table and using the 2004 U.S. population as million. Is your result from using the model an underestimate or an overestimate?

25. 3. Find P(17). What does it mean in this situation?
Use a Rational Model to Describe the Percentage of Quantity
Using a Rational Model to Make a Prediction
Example Continued
Formula
3. Find P(17). What does it mean in this situation?
Solution

26. Solution Continued Formula
Use a Rational Model to Describe the Percentage of Quantity
Using a Rational Model to Make a Prediction
Solution Continued
Formula
The model estimates that about 71.29% of Americans were Internet users in 2004
The actual percentage was
90.18% of Americans had internet service in 2007