Chapter 7 - Rational Expressions and Functions

1
Chapter 7 - Rational Expressions and Functions

• 7-1 Rational Expressions and Functions
• Multiplying and Dividing
• 7-2 Adding and Subtracting Rational Expressions
• 7-4 Equations with Rational Expressions and
  Graphs

2
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Defining rational expressions.
• A Rational Expression is the quotient of two
  polynomials with the denominator not equal to
  zero.
• Note Rational Expressions are the elements of
  the set
• Example 1 Rational Expressions

3
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Defining rational functions and describing their
  domain.
• A function that is defined by a rational
  expression is called a Rational Function and has
  the form
• Note The domain of a rational function contains
  all real numbers except those that make Q(x) 0.
  
• Example 2 Find all numbers that are in the
  domain of each rational function

4
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Defining rational functions and describing their
  domain.
• A function that is defined by a rational
  expression is called a Rational Function
• Example 3 Find all numbers that are in the
  domain of each rational function

5
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Defining rational functions and describing their
  domain.
• Example 4 Find all numbers that are in the
  domain of each rational function

6
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Writing rational expressions in lowest terms.
• Fundamental Property of of Rational Numbers
• If is a rational number and c is any
  nonzero number, then
• This is equivalent to multiplying by 1.
• Note A rational expression is a quotient of two
  polynomials. Since the value of a polynomial is a
  real number for every value of the variable for
  which it is defined, any statement that applies
  to rational numbers also applies to rational
  expressions.

7
7-1 Rational Expressions and Functions
Multiplying and Dividing

• Writing a rational expression in lowest terms.
• Step 1 Factor both the numerator and
  denominator to find their greatest common factor
  (GCF).
• Step 2 Apply the fundamental property.
• Example 4 Write each rational expression in
  lowest terms.

8
7-1 Rational Expressions and Functions
Multiplying and Dividing

• Writing a rational expression in lowest terms.
• Example 5 Write each rational expression in
  lowest terms.
• Note In general, if the numerator and the
  denominator are opposites, then the expression
  equals -1.

9
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Multiplying rational expressions.
• Step 1 Factor all numerators and denominators
  as completely as possible
• Step 2 Apply the fundamental property
• Step 3 Multiply remaining factors in the
  numerator and denominator, leaving the
  denominator in factored form.
• Step 4 Check to make sure the product is in
  lowest terms.
• Example 6 Multiply

10
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Multiplying rational expressions.
• Example 7 Multiply

11
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Finding reciprocals for rational expressions.
• To find the reciprocal of a non-zero rational
  expression, invert the rational expression.
• Example 8 Find all reciprocal
• Rational Expression Reciprocal

12
7-1 Rational Expressions and Functions Multipl
ying and Dividing

• Dividing rational expressions.
• To divide rational expressions, multiply the
  first by the reciprocal of the second.
• Example 9 Divide

13
7-2 Adding and Subtracting Rational Expressions

• Adding or subtracting rational expressions with
  the same common denominator.
• Step 1 If the denominators are the same, add or
  subtract the numerators and place the result over
  the common denominator.
• If the denominators are different, find the
  least common denominator. Write the rational
  expressions with the least common denominator and
  add or subtract numerators. Place the result over
  the common denominator.
• Step 2 Simplify by writing all answers in
  lowest terms
• Example 1 Add or subtract as indicated

14
7-2 Adding and Subtracting Rational Expressions

• Adding or subtracting rational expressions with
  different denominators.
• Step 1 If the denominators are different, find
  the least common denominator. Write the rational
  expressions with the least common denominator and
  add or subtract numerators. Place the result over
  the common denominator.
• Step 2 Simplify by writing all answers in
  lowest terms
• Finding the Least Common Denominator (LCD)
• Step 1 Factor each denominator.
• Step 2 Find the least common denominator. The
  LCD is the product of all different factors from
  each denominator, with each factor raised to the
  greatest power that occurs in any denominator.

15
7-2 Adding and Subtracting Rational Expressions

• Adding or subtracting rational expressions with
  different denominators.
• Step 1 If the denominators are different, find
  the least common denominator. Write the rational
  expressions with the least common denominator and
  add or subtract numerators. Place the result over
  the common denominator.
• Step 2 Simplify by writing all answers in
  lowest terms
• Example 2 Find the LCD for each pair of
  denominators

16
7-2 Adding and Subtracting Rational Expressions

• Adding or subtracting rational expressions with
  different denominators.
• Example 3 Add or subtract as indicated

17
7-2 Adding and Subtracting Rational Expressions

• Adding or subtracting rational expressions with
  different denominators.
• Example 4 Add or subtract as indicated

18
7-2 Adding and Subtracting Rational Expressions

• Example 5 Add or subtract as indicated

19
7-2 Adding and Subtracting Rational Expressions

• Example 6 Add or subtract as indicated

20
7-4 Equations with Rational Expressions and Graphs

• Determining the domain of a rational equation.
• The domain of a rational equation is the
  intersection (overlap)of the domains of the
  rational expressions in the equation.
• Example 1 Find the domain of the equation below

21
7-4 Equations with Rational Expressions and Graphs

• Solving rational equations.
• Multiply all terms of the equation by the least
  common denominator and solve. This technique may
  produce solutions that do not satisfy the
  original equation. All solutions have be checked
  in the original equation.
• Note This technique cannot be used on rational
  expressions.
• Example 2
• -3x 40 25 -3x -15
• x 5 Answer 5

22
7-4 Equations with Rational Expressions and Graphs

• Solving rational equations.
• Example 3

23
7-4 Equations with Rational Expressions and Graphs

• Solving rational equations. Example 4

24
7-4 Equations with Rational Expressions and Graphs

• Solving rational equations. Example 5

25
7-4 Equations with Rational Expressions and Graphs

• Recognizing the graph of a rational function.
• Because one or more values of x are excluded
  from the domain of most rational functions, their
  graphs are usually discontinuous. The graph
  produces a vertical asymptote at the value of x
  that is not allowed as part of the domain.
• Example 6

26
7-4 Equations with Rational Expressions and Graphs

• Recognizing the graph of a rational function.
• Because one or more values of x are excluded
  from the domain of most rational functions, their
  graphs are usually discontinuous. The graph
  produces a vertical asymptote at the value of x
  that is not allowed as part of the domain.
• Example 7