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SECTION 5.1 Simplifying Rational Expressions
Chapter 5 Rational Expressions, Equations, and Functions
Section 5.1: Simplifying Rational Expressions
Rational Expressions
Rational Expressions
Definition:
Simplifying:
MATH 1300 Fundamentals of Mathematics 297
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics298
SECTION 5.1 Simplifying Rational Expressions
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 299
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 1:
Solution:
Additional Example 2:
Solution:
University of Houston Department of Mathematics300
SECTION 5.1 Simplifying Rational Expressions
Additional Example 3:
Solution:
Additional Example 4:
MATH 1300 Fundamentals of Mathematics 301
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
University of Houston Department of Mathematics302
Exercise Set 5.1: Simplifying Rational Expressions
Simplify the following rational expressions. If the expression cannot be simplified any further, then simply rewrite the original expression.
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MATH 1300 Fundamentals of Mathematics 303
Exercise Set 5.1: Simplifying Rational Expressions
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University of Houston Department of Mathematics304
SECTION 5.2 Multiplying and Dividing Rational Expressions
Section 5.2: Multiplying and Dividing Rational Expressions
Multiplication and Division
Multiplication and Division
Multiplication of Rational Expressions:
To multiply two fractions, place the product of the numerators over the productof the denominators.
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 305
CHAPTER 5 Rational Expressions, Equations, and Functions
Division of Rational Expressions:
Example:
Solution:
University of Houston Department of Mathematics306
SECTION 5.2 Multiplying and Dividing Rational Expressions
Additional Example 1:
Solution:
Additional Example 2:
MATH 1300 Fundamentals of Mathematics 307
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
Additional Example 3:
University of Houston Department of Mathematics308
SECTION 5.2 Multiplying and Dividing Rational Expressions
Solution:
MATH 1300 Fundamentals of Mathematics 309
Exercise Set 5.2: Multiplying and Dividing Rational Expressions
Multiply the following rational expressions and simplify. No answers should contain negative exponents.
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University of Houston Department of Mathematics310
Exercise Set 5.2: Multiplying and Dividing Rational Expressions
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Divide the following rational expressions and simplify. No answers should contain negative exponents.
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MATH 1300 Fundamentals of Mathematics 311
CHAPTER 5 Rational Expressions, Equations, and Functions
Section 5.3: Adding and Subtracting Rational Expressions
Addition and Subtraction
Addition and Subtraction
Addition and Subtraction of Rational Expressions with Like Denominators:
Example: Perform the following operations. All results should be in simplified form.
University of Houston Department of Mathematics312
SECTION 5.3 Adding and Subtracting Rational Expressions
Solution:
Addition and Subtraction of Rational Expressions with Unlike Denominators:
Example:
MATH 1300 Fundamentals of Mathematics 313
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
Additional Example 1: Perform the following operations. All results should be in simplified form.
University of Houston Department of Mathematics314
SECTION 5.3 Adding and Subtracting Rational Expressions
Solution:
Additional Example 2: Perform the addition. Give the result in simplified form.
Solution:
MATH 1300 Fundamentals of Mathematics 315
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 3: Perform the subtraction. Give the result in simplified form.
Solution:
University of Houston Department of Mathematics316
SECTION 5.3 Adding and Subtracting Rational Expressions
Additional Example 4: Perform the subtraction. Give the result in simplified form.
Solution:
MATH 1300 Fundamentals of Mathematics 317
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 5: Perform the following operations. Give all results in simplified form.
University of Houston Department of Mathematics318
SECTION 5.3 Adding and Subtracting Rational Expressions
Solution:
MATH 1300 Fundamentals of Mathematics 319
Exercise Set 5.3: Adding and Subtracting Rational Expressions
Perform the indicated operations and simplify. (Whenever possible, write both the numerator and denominator of the answer in factored form.)
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University of Houston Department of Mathematics320
Exercise Set 5.3: Adding and Subtracting Rational Expressions
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MATH 1300 Fundamentals of Mathematics 321
CHAPTER 5 Rational Expressions, Equations, and Functions
Section 5.4: Complex Fractions
Simplifying Complex Fractions
Simplifying Complex Fractions
Definition:
Simplifying:
University of Houston Department of Mathematics322
SECTION 5.4 Complex Fractions
Example:
Solution:
Method 1:
MATH 1300 Fundamentals of Mathematics 323
CHAPTER 5 Rational Expressions, Equations, and Functions
Method 2:
University of Houston Department of Mathematics324
SECTION 5.4 Complex Fractions
Additional Example 1:
Solution:
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics 325
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 3:
Solution:
University of Houston Department of Mathematics326
SECTION 5.4 Complex Fractions
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics 327
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 5:
Solution:
University of Houston Department of Mathematics328
Exercise Set 5.4: Complex Fractions
Simplify the following. No answers should contain negative exponents.
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MATH 1300 Fundamentals of Mathematics 329
Exercise Set 5.4: Complex Fractions
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For each of the following expressions,(a) Rewrite the expression so that it contains
positive exponents rather than negative exponents.
(b) Simplify the expression.
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University of Houston Department of Mathematics330
Exercise Set 5.4: Complex Fractions
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MATH 1300 Fundamentals of Mathematics 331
CHAPTER 5 Rational Expressions, Equations, and Functions
Section 5.5: Solving Rational Equations
Rational Equations
Rational Equations
Definition of a Rational Equation:
Solving a Rational Equation:
Example:
Solution:
University of Houston Department of Mathematics332
SECTION 5.5 Solving Rational Equations
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 333
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics334
SECTION 5.5 Solving Rational Equations
Extraneous Solutions:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 335
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 1:
Solution:
University of Houston Department of Mathematics336
SECTION 5.5 Solving Rational Equations
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics 337
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 3:
Solution:
University of Houston Department of Mathematics338
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics 339
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 4:
Solution:
University of Houston Department of Mathematics340
Exercise Set 5.5: Solving Rational Equations
Solve the following. Remember to identify any extraneous solutions.
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MATH 1300 Fundamentals of Mathematics 341
Exercise Set 5.5: Solving Rational Equations
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University of Houston Department of Mathematics342
SECTION 5.6 Rational Functions
Section 5.6: Rational Functions
Working with Rational Functions
Working with Rational Functions
Definition of a Rational Function:
Domain of a Rational Function:
Example:
MATH 1300 Fundamentals of Mathematics 343
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
University of Houston Department of Mathematics344
SECTION 5.6 Rational Functions
Graph of a Rational Function:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 345
CHAPTER 5 Rational Expressions, Equations, and Functions
The graph of the function is shown below, labeled with the information from parts (b)-(d).
University of Houston Department of Mathematics346
SECTION 5.6 Rational Functions
Vertical Asymptotes:
MATH 1300 Fundamentals of Mathematics 347
CHAPTER 5 Rational Expressions, Equations, and Functions
Finding Vertical Asymptotes
Example:
Solution:
University of Houston Department of Mathematics348
SECTION 5.6 Rational Functions
Horizontal Asymptotes:
MATH 1300 Fundamentals of Mathematics 349
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 1:
Solution:
University of Houston Department of Mathematics350
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics 351
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 2:
Solution:
Additional Example 3:
University of Houston Department of Mathematics352
SECTION 5.6 Rational Functions
Solution:
MATH 1300 Fundamentals of Mathematics 353
CHAPTER 5 Rational Expressions, Equations, and Functions
University of Houston Department of Mathematics354
Exercise Set 5.6: Rational Functions
Find the indicated function values. If undefined, state “Undefined.”
1. If , find
(a) (b) (c)
2. If , find
(a) (b) (c)
3. If , find
(a) (b) (c)
4. If , find
(a) (b) (c)
5. If , find
(a) (b) (c)
6. If , find
(a) (b) (c)
7. If , find
(a) (b) (c)
8. If , find
(a) (b) (c)
9. If , find
(a) (b) (c)
10. If , find
(a) (b) (c)
The graph of each of the following functions has a horizontal asymptote at . (You will learn how to find horizontal asymptotes in a later mathematics course.) For each function,
(a) Find the domain of the function and express it as an inequality.
(b) Write the equation of the vertical asymptote(s) of the function.
(c) Find the x- and y-intercept(s) of the function, if they exist. If an intercept does not exist, state “None.”
(d) Find and .(e) Based on the features from (a)-(d), match the
function with its corresponding graph, using the choices (Graphs I-IV) below.
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MATH 1300 Fundamentals of Mathematics
Graph IV:
Graph I: Graph II:
Graph III:
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y
355
Exercise Set 5.6: Rational Functions
The graph of each of the following functions has a horizontal asymptote at . (You will learn how to find horizontal asymptotes in a later mathematics course.) For each function,
(a) Find the domain of the function and express it as an inequality.
(b) Write the equation of the vertical asymptote(s) of the function.
(c) Find the x- and y-intercept(s) of the function, if they exist. If an intercept does not exist, state “None.”
(d) Find and .(e) Based on the features from (a)-(d), match the
function with its corresponding graph, using the choices (Graphs I-IV) below.
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For each of the following functions,(a) Find the domain of the function and express it
as an inequality. Then write the domain of the function in interval notation.
(b) Write the equation of the vertical asymptote(s) of the function.
(c) Find the x- and y- intercept(s) of the function. If an intercept does not exist, state “None."
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University of Houston Department of Mathematics356
Graph IV:
Graph I: Graph II:
Graph III:
x
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x
y
x
y
x
y
Exercise Set 5.6: Rational Functions
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MATH 1300 Fundamentals of Mathematics 357
