[email protected] mth55_lec-29_fa08_sec_6-1_rational_fcn_mult-n-div.ppt 1 bruce mayer, pe...

[email protected] • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§6.1 Rational Fcn§6.1 Rational FcnMult & DivMult & Div



Review §Review §

Any QUESTIONS About• §5.7 → PolyNomial Equation

Applications

Any QUESTIONS About HomeWork• §5.7 → HW-17

5.7 MTH 55



Recall Rational FunctionRecall Rational Function

A rational function is a function, f(x), that is a quotient of two polynomials; i.e.

Where• where p(x) and q(x) are polynomials and

where q(x) is NOT the ZERO polynomial.

• The domain of f consists of all inputs x for which q(x) ≠ 0.



Rational FUNCTION ExampleRational FUNCTION Example

RATIONAL FUNCTION ≡ a function expressed in terms of rational expressions

Example Find f(3) for this Rational Function:

SOLUTION

2

2

3 7( ) ,

4

x xf x

x

2

2

3 7( )

4

x xf x

x

2

2

3 33( ) 7(3

3)

( ) 4f

9 9 7

9 5

11

4



Find the Domain of a Rational FcnFind the Domain of a Rational Fcn

1. Write an equation that sets the denominator of the rational function equal to 0.

2. Solve the denominator equation.

3. Exclude the value(s) found in step 2 from the function’s domain.



Example Example Find Domain Find Domain

Find the Domain for 3 2

6( ) .

5 4f y

y y y

SOLUTION

3 25 4 0y y y

2 5 4 0y y y

4 1 0y y y 0 or 4 0 or 1 0y y y

4 1y y

Set the denominator equal to 0.

Factor out the monomial GCF, y.

Use the zero products theorem.

The fcn is undefined for y = 0, −4, or −1, so the domain is {y|y −4, −1, 0}.

FOIL Factor the 2nd Degree polynomial



Example Example Find Domain Find Domain

Find the Domain for SOLUTION

• Find the values of x for which the denominator x2 – 6x + 8 = 0, then exclude those values from the domain.

The fcn is undefined for x = 2, or 4, so the domain is {x|x 2, 4}.• Interval Notation: (−∞,2)U(2,4)U(4,∞)

x 2 x 4 0

x 2 0 or x 4 0

x 2 or x 4



Example Example Graph Graph3

( )1

f xx

SOLUTION: x 1, so the graph has a vertical asymptote at x = 1. Find ordered pairs around the asymptote and then graph.

x 4 2 0 0.5

y 3/5 1 3 6

x 1.5 2 4 5

y 6 3 1 3/4



SimplifyingSimplifying Rational Rational Expressions and FunctionsExpressions and Functions As in arithmetic, rational expressions

are simplified by “removing”, or “Dividing Out”, a factor equal to 1.

example(2 1)( 5) (2 1) ( 5)

( 5( 7)( 5) )( 7)

x x x

x xx x

x

(2 1)

( 7)

x

x

removed the factor that equals 1

equals 1



Maintain DomainMaintain Domain

Because rational expressions often appear when we are writing functions, it is important that the function’s domain not be changed as a result of simplifying. For example, the Domain of the function given by

(2 1)( 5)( )

( 7)( 5)

x xF x

x x

is assumed to be all real numbers for which the denominator is NONzero




Thus for Rational Fcn: (2 1)( 5)

( )( 7)( 5)

x xF x

x x

In the previous example, we wrote F(x) in simplified form as

Domain of { 7, 5}.F x x x

(2 1).

( 7)

x

x

There is a serious problem with stating that these are equivalent; The Domains are NOT the same




Why (2 1)( 5)( )

( 7)( 5)

x xF x

x x

The domain of the function given by

Thus the domain of G includes 5, but the domain of F does not. This problem can be addressed by specifying

≠(2 1)

.( 7)

x

x

(2 1)( ) is assumed to be { 7}.

( 7)

xG x x x

x

(2 1)( 5) (2 1) 5.

( 7)( 5) ( 7)

x x xwith x

x x x



Example Example Maintain Domain Maintain Domain

Write this Fcn in Simplified form

SOLUTION: first factor the numerator and denominator, looking for the largest factor common to both. Once the greatest common factor is found, use it to write 1 and simplify as shown on the next slide

2

2

3 13 10( )

3 19 14

x xg x

x x



Example Example Maintain Domain Maintain Domain2

2

3 13 10( )

3 19 14

x xg x

x x

( 5)(3 2)

(3 2( 7) )

x

x

x

x

(3 2) ( 5)

(3 2) ( 7)

x x

x x

Note that the domain of g = {x | x 2/3 and x −7}by Factoring (see next)Factoring. The greatest common factor is (3x − 2).

Rewriting as a product of two rational expressions.

( 5)1

( 7)

x

x

For x 2/3, we have (3x − 2)/(3x − 2) = 1.



Example Example Maintain Domain Maintain Domain

Thus the simplified form of

23

( 5), .

( 7)

xx

x

Removing the factor 1. To keep the same domain, we specify that x 2/3.

2

2

3 13 10( )

3 19 14

x xg x

x x

23

5( ) , .

7

xg x with x

x



““Canceling” Confusion Canceling” Confusion

The operation of Canceling is a ShortHand for DIVISION between Multiplication Chains

Canceling can ONLY be done when we have PURE MULTIPLICATION CHAINS both ABOVE & BELOW the Division Bar`

11

34

7

214

7

214 22

x

x

xx

x

xx

3

7

73

7

2142

xx

xx

x

xx



Canceling CaveatCanceling Caveat “Canceling” is a shortcut often used for removing a

factor equal to 1 when working with fractions. Canceling removes multiplying factors equal to 1 in products. It cannot be done in sums or when adding expressions together. Simplifying the expression from the previous example might have been done faster as follows:

2

2

3 13 10 (3 2)( 5)

3 19 14 ( 7)(3 2)

x x x x

x x x x

( 5)

( 7)

x

x

When a factor that equals 1 is found, it is “canceled” as shown.

Removing a factor equal to 1.



Canceling CaveatCanceling Caveat

Caution! Canceling is often performed incorrectly:

Incorrect! Incorrect! Incorrect!

In each situation, the expressions canceled are not both factors. Factors are parts of products. For example, 5 is not a factor of the numerator 5x – 2. If you can’t factor, you can’t cancel! When in doubt, do NOT cancel!

7 5 2 2 3 17, ,

15 3 3

x x x

x x x

To check that these are not equivalent, substitute a number for x.



Simplifying Rational Expressions Simplifying Rational Expressions

1. Write the numerator and denominator in factored form.

2. Divide out all the common factors in the numerator and denominator; i.e., remove factors equal to ONE

3. Multiply the remaining factors in the numerator and the remaining factors in the denominator.



Example Example Simplify Simplify

SOLUTION:

2

2

3 9 12

6 30 24

x x

x x

Factor out the GCF.

2

2

3 9 12

6 30 24

x x

x x

2

2

3 3 4

6 5 4

x x

x x

3 4 1

6 4 1

x x

x x

4

2 4

x

x

Factor the polynomial factors.

Divide out common factors.

1432

143

xx

xx

1 and x



Multiply Rational ExpressionsMultiply Rational Expressions

The Product of Two Rational Expressions

To multiply rational expressions, multiply numerators and multiply denominators:

QS

PR

S

R

Q

P

Then factor and simplify the result if possible.



Example Example Multiplication Multiplication

Multiply and, if possible, simplify.

a) b)2

2

5 4 3

9 4

x x x

x x

46 5

10 6

a

a

SOLUTION a)4 46 (5)

10(

6 5

6 610 )

aa

a a

2 3 5

2 5 2 3

a a a a

a

2

3 a 5a a a 2 5 2 3 a

3

2

a

MULTIPLICATIONChains → Canceling OK



Example Example Multiplication Multiplication SOLUTION b)

2 2

2 2

5 4 3 ( 5 4)( 3)

9 4 ( 9)( 4)

x x x x x x

x x x x

2

2

5 4 3

9 4

x x x

x x

( 3)( 4)( 1)

()( 4)( 3 3)

x x

x xx

x

( 4)x

( 1) ( 3)x x

( 3) ( 3)x x ( 4)x

1

3

x

x




Example Example Multiply & Simplify Multiply & Simplify

Multiply and,if possible, simplify.

SOLUTION

( 4)( 5)

7(3)(3 1 )

7

)( 4

x x

x x

( 4)x

( 5) 7x

7 (3)(3 1) ( 4)x x ( 5)

3(3 1)

x

x

411321

720

4113

7

21

202

2

2

2

xx

xx

xx

xx

4113

7

21

202

2

xx

xx




Divide Rational ExpressionsDivide Rational Expressions

The Quotient of Two Rational Expressions

To divide by a rational expression, multiply by its reciprocal

WY

VZ

Y

Z

W

V

Z

Y

W

V

Then factor and, if possible, simplify.



Example Example Division Division

Divide and, if possible, simplify.

a) b)

SOLUTION

8

9

x

y

23

5

xx

x

9 9 8

8x x

y

y

72

xy

Multiplying by the reciprocal of the divisor

Multiplying rational expressions

a)

b) 5 23

52 3

1

x xx

x

x

x

( 3)( 5)

2

x x

x





SOLN

2 2

2 2

4 4 4

x x

x x x

2

2 2 2

2 2 2 4 4

4 4 4 4 2

x x x x x

x x x x x

( 2)( 2)

(

( 2)

( 22 ) ))( 2

x x

x

x

x x

( 2)x

( 2)x ( 2)

( 2)

x

x

( 2)x ( 2)x

2

2

x

x






2 2

2 2

3 2 5 6

5 4 10 24

x x x x

x x x x

SOLUTION ( 1)( 2)( 6)( 4)

( 4)( 1)( 2)( 3)

x x x x

x x x x

2 2 2 2

2 2 2 2

3 2 5 6 3 2 10 24

5 4 10 24 5 4 5 6

x x x x x x x x

x x x x x x x x

( 1)x

( 2)x ( 6) ( 4)x x

( 4)x ( 1)x ( 2)x ( 3)x

( 6)

( 3)

x

x






SOLUTION

2

2

3 4 1

25 6

x x x

x x

2 2

2 2

3 4 1 3 4 6

25 6 25 1

x x x x x x

x x x x

( 6)( 4)( 1)

()( 1)( 5 5)

x x

x xx

x

( 4) ( 1)x x

( 6)

( 5)( 5) ( 1)

x

x x x

( 4)( 6)

( 5)( 5)

x x

x x




Example Example Manufacturing Engr Manufacturing Engr

The function given by

gives the time, in hours, for two machines, working together to complete a job that the 1st machine could do alone in t hours and the 2nd machine could do in 3t − 2 hours. • How long will the two machines, working

together, require for the job if the first machine alone would take (a) 2 hours? (b) 5 hours?

23 2( )

4 2

t tM t

t



Example Example Manufacturing Engr Manufacturing Engr

SOLUTION23 2

( )4

2 2

22

2M

(a)

13

12 4 8 4or 1 hr

8 2 6 3

23 2( )

4

5 5

25

5M

1118

75 10 65or 3 hr

20 2 18

(b)



WhiteBoard WorkWhiteBoard Work

Problems From §6.1 Exercise Set• 22 (ppt), 26 (ppt), 114 , 16, 46, 66, 86

More Rational Division Since we are dividing fractions,

we multiply by the reciprocal Now, we follow the rule for

multiplication Factor and then cancel

Don't leave the numerator empty - put a one to hold the place.



P6.1-22 P6.1-22 Rational fcn Graph Rational fcn Graph

Describe end-behavior of Graph at Far-Right

ANS: As x becomes large y = f(x) approaches, but never reaches, a value of 3




What is the Eqn for the Horizontal Asymptote:

ANS: y = f(x) approaches, but never reaches, a value of 3, to the Asymptote eqn

y = 3




List 2 real No.s that are NOT function values of f

ANS: y = f(x) does not have a graph between y > 0 and y < 3. Thus two values for which there is NO f(x):• y = 1 or y = 2



P6.1-114 P6.1-114 Smoking Diseases Smoking Diseases

Find P(9). Describe Meaning. ID pt on Graph

ANS: An incidence ratio of 9 indicates that 88.9% of Lung Cancer deaths are associated with Cigarette smoking

988

9

800

9

191009 .

P



P6.1-114 P6.1-114 Smoking Diseases Smoking Diseases

ID P(9) on Graph

988.



All Done for TodayAll Done for Today

Diophantus of

Alexandria

The FIRST Algebraist• In the 3rd century, the

Greek mathematician Diophantus of Alexandria wrote his book Arithmetica. Of the 13 parts originally written, only six still survive, but they provide the earliest record of an attempt to use symbols to represent unknown quantities.



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



Graph Graph yy = | = |xx||

Make T-tablex y = |x |

-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

x

y

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



x

y

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls

xy

[email protected] mth55_lec-29_fa08_sec_6-1_rational_fcn_mult-n-div.ppt 1 bruce mayer, pe...

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