Simplification of Rational Expressions and Functions


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1 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work in this chapter will epand our eperience with algebraic epressions to include algebraic fractions or rational epressions. We consider the four basic operations of addition, subtraction, multiplication, and division in the net sections. Fortunatel, ou will observe man parallels to our previous work with arithmetic fractions. First, let s define what we mean b a rational epression. Recall that a rational number is the ratio of two integers. Similarl, a rational epression can be written as the ratio of two polnomials, in which the denominator cannot have the value 0. Definitions: Rational Epression NOTE The word rational comes from ratio. A rational epression is the ratio of two polnomials. It can be written as P Q in which P and Q are polnomials and Q cannot have the value 0. The epressions and are all rational epressions. The restriction that the denominator of the epressions not be 0 means that certain values for the variable ma have to be ecluded because division b 0 is undefined. Eample 1 Precluding Division b Zero (a) For what values of is the following epression undefined? NOTE A fraction is undefined when its denominator is equal to 0. NOTE Note that when 5, 5 becomes or , 0. 5 To answer this question, we must find where the denominator is 0. Set 5 0 or 5 The epression is undefined for
2 SIMPLIFICATION OF RATIONAL EXPRESSIONS AND FUNCTIONS SECTION (b) For what values of is the following epression undefined? 3 5 Again, set the denominator equal to 0: 5 0 or 5 The epression 3 5 is undefined for 5. CHECK YOURSELF 1 For what values of the variable are the following epressions undefined? (a) 1 r 7 (b) Generall, we want to write rational epressions in the simplest possible form. To begin our discussion of simplifing rational epressions, let s review for a moment. As we pointed out previousl, there are man parallels to our work with arithmetic fractions. Recall that so 3 5 and 6 10 name equivalent fractions. In a similar fashion, so and 2 3 name equivalent fractions. We can alwas multipl or divide the numerator and denominator of a fraction b the same nonzero number. The same pattern is true in algebra. Rules and Properties: For polnomials P, Q, and R, P Q PR when Q 0 and R 0 QR Fundamental Principle of Rational Epressions
3 488 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS NOTE In fact, ou will see that most of the methods in this chapter depend on factoring polnomials. This principle can be used in two was. We can multipl or divide the numerator and denominator of a rational epression b the same nonzero polnomial. The result will alwas be an epression that is equivalent to the original one. In simplifing arithmetic fractions, we used this principle to divide the numerator and denominator b all common factors. With arithmetic fractions, those common factors are generall eas to recognize. Given rational epressions in which the numerator and denominator are polnomials, we must determine those factors as our first step. The most important tools for simplifing epressions are the factoring techniques in Chapter 6. NOTE We find the common factors 4,, and in the numerator and denominator. We divide the numerator and denominator b the common factor 4. Note that NOTE We have divided the numerator and denominator b the common factor 2. Again note that Eample 2 Simplifing Rational Epressions Simplif each rational epression. Assume denominators are not 0. (a) (b) ( 2) ( 2)( 2) Factor the numerator and the denominator. We can now divide the numerator and denominator b the common factor 2: 3( 2) ( 2)( 2) 3 2 CAUTION Pick an value other than 0 for the variable, and substitute. You will quickl see that and the rational epression is in simplest form. Be Careful! Given the epression 2 3 students are often tempted to divide b variable, as in 2 3 This is not a valid operation. We can onl divide b common factors, and in the epression above, the variable is a term in both the numerator and the denominator. The numerator and denominator of a rational epression must be factored before common factors are divided out. Therefore, is in its simplest possible form. CHECK YOURSELF 2 Simplif each epression. 36a 3 b (a) (b) ab
4 SIMPLIFICATION OF RATIONAL EXPRESSIONS AND FUNCTIONS SECTION The same techniques are used when trinomials need to be factored. Eample 3 further illustrates the simplification of rational epressions. Eample 3 Simplifing Rational Epressions NOTE Divide b the common factor 1, using the fact that when 1 NOTE In part (c) we factor b grouping in the numerator and use the sum of cubes in the denominator. Note that ( 2) 3( 2) ( 2)( 2 3) Simplif each rational epression. (a) (b) (2 1) 5( 1)( 1) ( 5)( 1) ( 2)(2 3) ( 1)(2 3) 2 1 (c) ( 1) 5 ( 2)( 2 3) ( 2)( 2 2 4) CHECK YOURSELF 3 Simplif each rational epression (a) (b) Simplifing certain algebraic epressions involves recognizing a particular pattern. Verif for ourself that 3 9 (9 3) In general, it is true that a b a (b) b a 1(b a) or, b dividing the left and right sides of the equation b b a, NOTE Notice that a b a b 1 but a b b a 1 when a b. NOTE From the margin note above, we get a b (b a) 1 b a b a Eample 4 makes use of this result. Eample 4 Simplifing Rational Epressions Simplif each rational epression (a) 4 2 2( 2) (2 )(2 ) 2(1) 2 2 2
5 490 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS (3 )(3 ) (b) ( 5)( 3) (3 )(1) CHECK YOURSELF 4 Simplif each rational epression (a) (b) The following algorithm summarizes our work with simplifing rational epressions. Step b Step: Simplifing Rational Epressions Step 1 Step 2 Step 3 Completel factor both the numerator and denominator of the epression. Divide the numerator and denominator b all common factors. The resulting epression will be in simplest form (or in lowest terms). To identif rational functions, we begin with a definition. Definitions: Rational Function A rational function is a function that is defined b a rational epression. It can be written as f() P Q in which P and Q are polnomials and Q() 0 for all. Eample 5 Identifing Rational Functions NOTE Recall from Chapter 6 that there are no square roots of variables in a polnomial. Which of the following are rational functions? (a) f() (b) f() (c) f() This is a rational function; it could be written over the denominator 1, and 1 is a polnomial. This is a rational function; it is the ratio of two polnomials. This is not a rational function; it is not the ratio of two polnomials.
6 SIMPLIFICATION OF RATIONAL EXPRESSIONS AND FUNCTIONS SECTION CHECK YOURSELF 5 Which of the following are rational functions? (a) f() (c) f() (b) f() When we simplif a rational function, it is important that we note the values that need to be ecluded, particularl when we are tring to draw the graph of a function. The set of ordered pairs of the simplified function will be eactl the same as the set of ordered pairs of the original function. If an ecluded value for ields ordered pair (, f()), that ordered pair represents a hole in the graph. These holes are breaks in the curve. We use an open circle to designate them on a graph. Eample 6 Simplifing a Rational Function Given the function f() 1 complete the following. NOTE Notice that f(1) is undefined. (a) Simplif the rational epression on the right ( 1)( 1) ( 1) ( 1) 1 (b) Rewrite the function in simplified form. f() 1 1 (c) Find the ordered pair associated with the hole in the graph of the original function. Plugging 1 into the simplified function ields the ordered pair (1, 0). This represents the hole in the graph of the function f() CHECK YOURSELF 6 Given the function f() complete the following. (a) Rewrite the function in simplified form. (b) Find the ordered pair associated with the hole in the graph of the original function.
7 492 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS Certain rational functions can be graphed as a line with a hole in it. One such function is eamined in Eample 7. Eample 7 Graphing a Rational Function Graph the following function f() 1 From Eample 6, we know that Therefore, f() 1 1 Because an value of 1 results in division b 0, there can be no point on the graph with an value of 1. The graph will be the graph of the line f() 1, with an open circle at the point (1, 0). CHECK YOURSELF 7 Graph the function f() CHECK YOURSELF ANSWERS 4a (a) r 7; (b) 9 2(a) ; (b) 3. (a) ; (b) 2 b (a) ; (b) (a) A rational function; (b) not a rational function; and (c) a rational function 6. (a) f() 2, 0; (b) (0, 2) 7.
8 Name 7.1 Eercises Section Date In eercises 1 to 12, for what values of the variable is each rational epression undefined? ANSWERS ( 1) In eercises 13 to 16, evaluate each epression, using a calculator for for for for In eercises 17 to 48, simplif each epression. Assume the denominators are not a 2 b 3 24a 4 b p 5 q 3 r p 3 q 5 r a 5 b 3 c 2 84a 2 bc
9 ANSWERS m 2 11m 21 3b 2 14b m 2 9 b z 5z z 15z r 2 rs 6s 2 a r 3 8s 3 a 2 5a 6 a 2 9b 2 a 2 8ab 15b c 4 16 c 2 3c cd 3c 5d d 2 7d m m In eercises 49 to 54, identif which functions are rational functions. 49. f() f() f() 52. f() f() f() For the given functions in eercises 55 to 60, (a) rewrite the function in simplified form, and (b) find the ordered pair associated with the hole in the graph of the original function f() 56. f() f() 58. f() f() 60. f() ( 2) 7( 3) 494
10 ANSWERS In eercises 61 to 66, graph the rational functions. Indicate the coordinates of the hole in the graph f() 62. f() 63. f() f() 65. f() 66. f() Eplain wh the following statement is false. 6m 2 2m 2m 6m State and eplain the Fundamental Principle of Rational Epressions The rational epression can be simplified to 2. Is this 2 reduction true for all values of? Eplain. 70. What is meant b a rational epression in lowest terms? In eercises 71 to 76, simplif. 2( h) ( h) 72. 3( h) 3 (3 3) 73. ( h) 74. ( h) ( h) 3( h) (3) ( h) 2( h) 5 (2 5) ( h) ( h) 3 3 ( h) 495
11 ANSWERS P() Given f(), if the graphs of P() and Q() intersect at (a, 0), then a is a factor Q() of both P() and Q(). Use a graphing calculator to find the common factor for the epressions in eercises 77 and (a) (b) (c) (d) (e) f() 78. f() Revenue. The total revenue from the sale of a popular video is approimated b the rational function R() (f) in which is the number of months since the video has been released and R() is the total revenue in hundreds of dollars. (a) Find the total revenue generated b the end of the first month. (b) Find the total revenue generated b the end of the second month. (c) Find the total revenue generated b the end of the third month. (d) Find the revenue in the second month onl. 80. Cost. A compan has a setup cost of $3500 for the production of a new product. The cost to produce a single unit is $8.75. (a) Define a rational function that gives the average cost per unit when units are produced. (b) Find the average cost when 50 units are produced. 81. Besides holes, we sometimes encounter a different sort of break in the graph of a rational function. Consider the rational function f() 1 3 (a) For what value(s) of is the function undefined? (b) Complete the following table f() 496
12 ANSWERS (c) What do ou observe concerning f() if is chosen close to 3 (but slightl larger than 3)? (d) Complete the table. f() (a) (b) (c) (d) (e) (f) (e) What do ou observe concerning f() if is chosen close to 3 (but slightl smaller than 3)? (f) Graph the function on our graphing calculator. Describe the behavior of the graph of f() near Consider the rational function f() 1 2 (a) For what value(s) of is the function undefined? (b) Complete the following table f() (c) What do ou observe concerning f() if is chosen close to 2? (but slightl smaller than 2)? (d) Complete the following table f() (e) What do ou observe concerning f() if is chosen close to 2 (but slightl larger than 2)? (f) Graph the function on our graphing calculator. Describe the behavior of the graph near Age Ratios. Your friend has a 4earold cousin, Am, who has a 9earold brother. The ounger child is upset because not onl does her brother refuse to let her pla with him and his friends, he teases her because she can never catch up in age! You eplain to the child that as she gets older, this age difference will not seem like such a big deal. Write an epression for the ratio of the ounger child s age to her brother s age. 497
13 1. What happens to the ratio as the children grow older? 2. Draw a graph of the ratio as a function of Am s age to show how this ratio changes. 3. Assume that Am and her brother live to be 100 and 105 ears old, respectivel. What will the ratio between their ages be? 4. Draw a graph of the ratio of Am s brother s age to Am s age. Does this graph have anthing in common with the first graph? Eplain. 5. Write a short eplanation for Am (who is onl 4, remember!) of our conclusions about how their age ratios change over time. Answers Never undefined a 3 b c z b z (a 2 9)(a 3) a m Rational 51. Rational 53. Not rational 55. (a) f() 2; (b) (1, 3) 57. (a) f() 3 1; (b) (2, 7) (a) f() ; (b) (2, 0) (1, 2) (2, 6) (1, 2) h (a) $3000; (b) $9231; (c) $15,000; (d) $ (a) 3; (b) 1, 10, 100, 1000, 10,000; (c) ; (d) 1, 10, 100, 1000, 10,
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