Chapter 6 Rational Expressions, Functions, and Equations.

Chapter 6Rational Expressions,

Functions, and Equations

§ 6.1

Rational Expressions and Functions: Multiplying and Dividing

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.1

Rational Expressions

A rational expression consists of a polynomial divided by a nonzero polynomial (denominator cannot be equal to 0).

A rational function is a function defined by a formula that is a rational expression. For example, the following is a rational function:

x

xxf

100

130



EXAMPLEEXAMPLE

The rational function

models the cost, f (x) in millions of dollars, to inoculate x% of the population against a particular strain of flu. The graph of the rational function is shown. Use the function’s equation to solve the following problem.

Find and interpret f (60). Identify your solution as a point on the graph.

x

xxf

100

130

p 393



CONTINUECONTINUEDD

0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100



SOLUTIONSOLUTION

We use substitution to evaluate a rational function, just as we did to evaluate other functions in Chapter 2.

CONTINUECONTINUEDD

x

xxf

100

130 This is the given rational function.

60100

6013060

f

Replace each occurrence of x with 60.

19540

7800 Perform the indicated

operations.



Thus, f (60) = 195. This means that the cost to inoculate 60% of the population against a particular strain of the flu is $195 million. The figure below illustrates the solution by the point (60,195) on the graph of the rational function.

CONTINUECONTINUEDD

0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100

(60,195)


Rational Expressions - Domain

EXAMPLEEXAMPLE

Find the domain of f if

.3613

32

xx

xxf

The domain of f is the set of all real numbers except those for which the denominator is zero. We can identify such numbers by setting the denominator equal to zero and solving for x.

SOLUTIONSOLUTION

036132 xx Set the denominator equal to 0.

094 xx Factor.p 393



Because 4 and 9 make the denominator zero, these are the values to exclude. Thus,

9 and 4 andnumber real a is | ofDomain xxxxf

Set each factor equal to 0.04 x

Solve the resulting equations.

CONTINUECONTINUEDD

09 xor4x 9x

. ,99,4,4- ofDomain for



CONTINUECONTINUEDD . ,99,4,4- ofDomain f

In this example, we excluded 4 and 9 from the domain. Unlike the graph of a polynomial which is continuous, this graph has two breaks in it – one at each of the excluded values. Since x cannot be 4 or 9, there is not a function value corresponding to either of those x values. At 4 and at 9, there will be dashed vertical lines called vertical asymptotes. The graph of the function will approach these vertical lines on each side as the x values draw closer and closer to each of them, but will not touch (cross) the vertical lines. The lines x = 4 and x = 9 each represent vertical asymptotes for this particular function.

.)9)(4(

3

xx

xxf

p 395



AsymptotesVertical

AsymptotesA vertical line that the graph of a function approaches, but does not touch.

Horizontal Asymptotes

A horizontal line that the graph of a function approaches as x gets very large or very small. The graph of a function may touch/cross its horizontal asymptote.

Simplifying Rational Expressions1) Factor the numerator and the denominator completely.

2) Divide both the numerator and the denominator by any common factors.

p 395



Check Point 2Check Point 2

Find the domain of f if .352

52

xx

xxf

SOLUTIONSOLUTION

0352 2 xx Set the denominator equal to 0.

0312 xx Factor.

p 394

012 x

Solve the resulting equations.

03 x

2

1x 3x

Set each factor equal to 0.or

2

1 and 3 andnumber real a is | ofDomain xxxxf

. ,2

1

2

1,3,-3- ofDomain

for


Simplifying Rational Expressions

EXAMPLEEXAMPLE

Simplify: .32

12

xx

x

SOLUTIONSOLUTION

Factor the numerator and denominator.

13

11

32

12

xx

x

xx

x

Divide out the common factor, x + 1.

13

11

xx

x

3

1

xSimplify.




Simplify: .2

1072

x

xx

SOLUTIONSOLUTION


21

25

2

1072

x

xx

x

xx

Divide out the common factor, x + 1.

21

25

x

xx

5x Simplify.

p 397



EXAMPLEEXAMPLE

Simplify: .9

1272

2

x

xx

SOLUTIONSOLUTION


Rewrite 3 – x as (-1)(-3 + x).

xx

xx

x

xx

33

43

9

1272

2

xx

xx

313

43

313

43

xx

xxRewrite -3 + x as x – 3.



Divide out the common factor, x – 3.

Simplify.

313

43

xx

xx

CONTINUECONTINUEDD

13

4

x

x

Do Check 4a and 4b on page 397


Multiplying Rational Expressions

Multiplying Rational Expressions1) Factor all numerators and denominators completely.

2) Divide numerators and denominators by common factors.

3) Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.

p 398



EXAMPLEEXAMPLE

Multiply: .6

23

43

23222

22

22

22

yxyx

yxyx

yxyx

yxyx

SOLUTIONSOLUTION

Factor the numerators and denominators completely.

Divide numerators and denominators by common factors.

yxyx

yxyx

yxyx

yxyx

32

3

3

22

22

22

22

22

6

23

43

232

yxyx

yxyx

yxyx

yxyx

This is the original expression.

yxyx

yxyx

yxyx

yxyx

32

3

3

22



Multiply the remaining factors in the numerators and in the denominators.

yxyx

yxyx

33

32

CONTINUECONTINUEDD

Note that when simplifying rational expressions or multiplying rational expressions, we just used factoring.

With one additional step that is provided in the followingDefinition for Division, division of rational expressions promises to be just as straightforward.



EXAMPLEEXAMPLE

Multiply: .4

3

9

2822 yy

y

y

y

SOLUTIONSOLUTION

Factor the numerators and denominators completely.

Divide numerators and denominators by common factors. Because 3 – y and y -3 are opposites, their quotient is -1.

This is the original expression.yy

y

y

y

22 4

3

9

28

14

3

33

142

yy

y

yy

y

14

)3(1

33

142

yy

y

yy

y



Now you may multiply the remaining factors in the numerators and in the denominators.

yy 3

2

CONTINUECONTINUEDD

or yy 3

2




Multiply: .16

214

7

42

2

x

xx

x

x

4

3

x

x

pages 398-399




Multiply: .49

443

36

842

2

2

x

xx

xx

x

233

24

xx

x 233

24

xx

x

pages 398-399


Dividing Rational Expressions

Simplifying Rational Expressions with Opposite Factors in the Numerator and Denominator

The quotient of two polynomials that have opposite signs and are additive inverses is -1.

Dividing Rational ExpressionsIf P, Q, R, and S are polynomials, where then,0 and ,0 ,0 SRQ

.QR

PS

R

S

Q

P

S

R

Q

P

Change division to multiplication.

Replace with its reciprocal by interchanging its numerator and denominator.

S

R



EXAMPLEEXAMPLE

Divide: .352

32

12 22

22

2

2

yxyx

yxyx

xx

yxy

SOLUTIONSOLUTION

Invert the divisor and multiply.

Factor.

This is the original expression.22

22

2

2

352

32

12 yxyx

yxyx

xx

yxy

22

22

2

2

32

352

12 yxyx

yxyx

xx

yxy

yxyx

yxyx

x

yxy

32

32

1 2



Divide numerators and denominators by common factors.

yxyx

yxyx

x

yxy

32

32

1 2

CONTINUECONTINUEDD

21

x

yxy Multiply the remaining factors in the numerators and in the denominators.



Check Point 7aCheck Point 7a

Divide: .9

73499 2

x

x

739 x



Check Point 7bCheck Point 7b

Divide: .6

2410

5

122

22

xx

xx

x

xx

5

3

x

DONE

Chapter 6 Rational Expressions, Functions, and Equations.

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