Chapter 7 Rational Functions - College of the Redwoods

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618 Chapter 7 Rational Functions 13. D = {x : x ≠ 3}, R = {y : y ≠ −4} y 10 x=3 y=−4 x 10 15. Vertical asymptote: x = −1 17. Vertical asymptote: x = −2 19. Vertical asymptote: x = −5 21. Vertical asymptote: x = −8 23. Horizontal asymptote: y = 9 25. Horizontal asymptote: y = −1 27. Horizontal asymptote: y = −9 29. Horizontal asymptote: y = −4 31. Domain = {x : x ≠ −5} 33. Domain = {x : x ≠ 5} 35. Domain = {x : x ≠ −7} 37. Domain = {x : x ≠ −2} 39. Range = {y : y ≠ 8} 41. Range = {y : y ≠ −5} 43. Range = {y : y ≠ 5} 45. Range = {y : y ≠ −2} Version: Fall 2007
Section 7.2 Reducing Rational Functions 619 7.2 Reducing Rational Functions The goal of this section is to learn how to reduce a rational expression to “lowest terms.” Of course, that means that we will have to understand what is meant by the phrase “lowest terms.” With that thought in mind, we begin with a discussion of the greatest common divisor of a pair of integers. First, we define what we mean by “divisibility.” Definition 1. Suppose that we have a pair of integers a and b. We say that “a is a divisor of b,” or “a divides b” if and only if there is another integer k so that b = ak. Another way of saying the same thing is to say that a divides b if, upon dividing b by a, the remainder is zero. Let’s look at an example. ◮ Example 2. What are the divisors of 12? Because 12 = 1 × 12, both 1 and 12 are divisors 6 of 12. Because 12 = 2 × 6, both 2 and 6 are divisors of 12. Finally, because 12 = 3 × 4, both 3 and 4 are divisors of 12. If we list them in ascending order, the divisors of 12 are 1, 2, 3, 4, 6, and 12. Let’s look at another example. ◮ Example 3. What are the divisors of 18? Because 18 = 1 × 18, both 1 and 18 are divisors of 18. Similarly, 18 = 2 × 9 and 18 = 3 × 6, so in ascending order, the divisors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common divisor of two or more integers is the largest divisor the integers share in common. An example should make this clear. ◮ Example 4. What is the greatest common divisor of 12 and 18? In Example 2 and Example 3, we saw the following. Divisors of 12 : 1 , 2 , 3 , 4, 6 , 12 Divisors of 18 : 1 , 2 , 3 , 6 , 9, 18 5 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 6 The word “divisor” and the word “factor” are synonymous. Version: Fall 2007
Page 1 and 2: 7 Rational Functions In this chapte
Page 3 and 4: Section 7.1 Introducing Rational Fu
Page 17: Section 7.1 Introducing Rational Fu
Page 21 and 22: Section 7.2 Reducing Rational Funct
Page 39 and 40: Section 7.3 Graphing Rational Funct
Page 61 and 62: Section 7.4 Products and Quotients
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Section 7.4 Products and Quotients
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Section 7.5 Sums and Differences of
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Section 7.6 Complex Fractions 695 7
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Section 7.9 Index 747 7.9 Index a a
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Chapter 7 Rational Functions - College of the Redwoods

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