Chapter 12 Rational Functions

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

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.

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

Domain of a Rational Function Definition of a Rational Function Example Find the domain of the function. 1. 2. 1. 3 is not in the domain because Solution Domain: { x | x ≠ 3 }

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

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. 1. 2. Example

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 ≠ }  

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.

Simplifying Rational Expressions Example of Simplifying Rational Expressions Example Simplifying

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

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.

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

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 }

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

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

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

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

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

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

Quotient Function Quotient Functions Solution

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

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 1990. In Exercise 9 of Section 7.7, you modeled the U.S. population. A reasonable model is U(t) = 0.0068t2 + 2.58t + 251.7, where U(t) is the U.S. population (in millions) at t years since 1990.

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 1990. Find an equation of P. 2. Use P to estimate the percentage of Americans who were Internet users in 2004. Then compute the actual percentage by referring to the table and using the 2004 U.S. population as 293.7 million. Is your result from using the model an underestimate or an overestimate?

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

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