functionsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter... · 2018-08-31 · lecture 6:...

Chapter 2: Rational Functions

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Chapter 2: Rational FunctionsLecture 6: Basic Concepts

Lecture 7: Solving Rational EquationsLecture 8: Solving Rational InequalitiesLecture 9: Asymptotes of Rational

FunctionsLecture 10: Graphing Rational

Functions

Family Activity 1:

Constructing KWL ChartSHMth1: General Mathematics


Management (ABM

Mr. Migo M. Mendoza

The Grading SystemCriteria Percentage

Content 18 points

Organization of Ideas 5 points

Communication Skills 5 points

Presentation and

Aesthetic

Consideration

3 points

Behavior 4 points

What to do?•Answer the question:

“What is a Rational Function?”

by constructing a “KWL” (Know, Want to

Know, Learned) Chart. Afterwards, share your answer to the class. Please be guided

that this is a time pressure family activity.

The KWL ChartKnow Want to

KnowLearned

1.

2.

3.

1.

2.

3.

Lecture 6: Basic Concepts

in Rational Functions



Management (ABM

Mr. Migo M. Mendoza

Rational Number (ℚ)It is a number that can be

written as a fraction or ratio and whose numerators and

denominators are integers provided that the denominator

is not equal to 0.

The Definition of Rational FunctionIf we let P(x) and Q(x) be two

polynomials, then a function of the form:

is called a RATIONAL FUNCTION.

)(

)()(

xQ

xPxf

The Domain of the Rational Function

The domain of f(x) is the

set of real numbers x except those for which

Q(x) = 0.

Take Note:Since division by zero is

impossible, a rational function has a DISCONTINUITY

whenever its denominator is zero.

Did you know?

The denominator of a rational function cannot be zero. Any

value of the variable that would make the denominator zero is

NOT PERMISSIBLE.

Take Note:

The domain of a rational function is the set of all real numbers, except those that

make the denominator zero.

Some Examples of Rational Function:

2;2

23)(

x

x

xxf


3

2;

23

14)(

2

x

x

xxxf


3;3

1)(

x

xxf

A Short Review on Rational Functions:

Find the domain of the rational function:

)3()(

xx

xxr

Final Answer:

The domain of r(x) is the set of all real

numbers, except those that make the denominator zero. Thus,

}30{ andxxxD

A Short Review on Rational Functions:

Find the domain of the rational function:

82

54)(

2

2

xx

xxxR

Final Answer:

The domain of r(x) is the set of all real

numbers, except those that make the denominator zero. Thus,

}24{ andxxxD

Rational Function

in Real WorldSHMth1: General Mathematics


Management (ABM

Mr. Migo M. Mendoza

In Pharmacology…We use rational function to determine

the medicine concentration after a period of time. Say:

3.301.0

5)(

2

t

ttC

In Biology…A biologist discovered a formula to

determine how your blood brings oxygen to the rest of the body—the HEMOGLOBIN.

d

n

n

Kl

l

In Investment…Rational function is used to determine exact

and ordinary interest which is use in Banker’s Rule.

365

Pr tIe

360

Pr tIo

In Consumer Loan…Rational function is used to determine the

borrower’s equal payment at equal interval (annuity).

1)1(

n

n

i

iSR

Lecture 7: Solving Rational Equations



Management (ABM

Mr. Migo M. Mendoza

Rational Expression

A rational expression is an expression that can

be written as a ratio of two polynomials.

Family Activity 2:

Am I Rational Expression?SHMth1: General Mathematics


Management (ABM

Mr. Migo M. Mendoza

The Grading SystemCriteria Percentage

Correctness 10 points

Justification/

Reasoning6 points

Communication

Skills5 points

Behavior 4 points

Am I Rational Expression?

Using the definition of a rational expression, tell why the

following is or not a rational expression. Have a sound

justification.

Am I Rational Expression?

2

34)(

3

1)(

3

333)(

2

2

2

xxc

xb

x

xxa

2

2

1

)(

1

1)(

3

x

xe

x

xd

To sum it up…

In simplest manner, a rational expression can be described as a

function where either the numerator, denominator, or

both have a variable on it.

Lecture 7: Solving Rational Equations



Management (ABM

Mr. Migo M. Mendoza

Something to think about…

What is your idea of a rational equation?

Example 46:Solve the rational equation:

3

1

2

1

6

5

x

Something to think about…

How to solve rational equations?

Solving Rational EquationsTo solve a rational equation,

we multiply each term of the equation by the least

common denominator (LCD) of any fractions.

Solving Rational Equations

The resulting equation should be equivalent to the original

equation and be cleared of all fractions as long as we do not

multiply by zero.

Steps in Solving an Equation Containing Rational Equations

Step 1:Determine the LCD of

all the denominators.


Step 2:Multiply each term of the equation by the LCD.


Step 3:Solve the resulting

equation.


Step 4:Check your answer by substituting it into

the original equation. Exclude from the solution any value that would make the LCD equal to zero. Such value is called

EXTRANEOUS SOLUTION.

Final Answer:

Hence,

is the solution of the given equation.

1x


31

2

13

4

xx

Final Answer:

Hence,

is the root of the given equation.

6x


2

12

2

3

xxx

Final Answer:Hence,

is the only root of the given equation.

1x


212

3 x

x

x

Final Answer:The roots of the given rational

equation are:

.36 andxx


43

1

20193

73

5

12

xxx

x

x

Final Answer:The root of the given rational

equation is:

6x

Performance Task 6:

Please download, print

and answer the “Let’s

Practice 6.” Kindly work

independently.

Lecture 8: Solving Rational InequalitiesSHMth1: General Mathematics


Management (ABM

Mr. Migo M. Mendoza

Rational Inequalities

An inequality that contains rational expressions is

referred to as RATIONAL INEQUALITIES.

Rational Inequalities

It is a rational equation that contains

an inequality.

Example 50:

Solve the rational inequality, then, graph its solution set:

4

1

1

2

xx

Step 1: Solving Rational Inequalities Determine the critical numbers for

f(x) by establishing the zeros of f(x)

and excluded values for f(x). We

can solve for the zeros of f(x) using

the numerator of the rational

function.

Step 2: Solving Rational Inequalities

Plot the critical numbers

in the number line into

intervals and create a table

for test of values of x.

Test of Values

Intervals

Test of

Value x

f(x)

Sign of f(x)

Rational Inequality Theorem 1: If the rational inequality is of the

form f(x) > 0 or f(x) ≥ 0, then all

of the intervals with the positive sign are solutions. Also, the

zeros of f(x) are part of the

solution if f(x) ≥ 0.

Final Answer:

The solution is the interval notation:

.149| orxxx

Example 51:

Solve the rational inequality, then, graph its solution set:

12

532

x

xx

Step 1: Solving Rational Inequalities Determine the critical numbers for

f(x) by establishing the zeros of f(x)

and excluded values for f(x). We

can solve for the zeros of f(x) using

the numerator of the rational

function.

Step 2: Solving Rational Inequalities

Plot the critical numbers

in the number line into

intervals and create a table

for test of values of x.

Test of Values

Intervals

Test of

Value x

f(x)

Sign of f(x)

Rational Inequality Theorem 2: If the rational inequality is of the

form f(x) < 0 or f(x) ≤ 0, then all

of the intervals with the negative sign are solutions. Also, the

zeros of f(x) are part of the

solution if f(x) ≤ 0.

Final Answer:

Since the rational inequality is

of the form f(x) < 0, then the

solution is:

.213| xorxx

Performance Task 7:




independently.

Lecture 9: Asymptotes of Rational Function



Management (ABM

Mr. Migo M. Mendoza

Did you know?

There are three (3)

saddest love stories in

Mathematics…

The Painful Asymptote

There are people who

may get closer and

closer to one another,

but will never be

together.

The Painful Parallel

You may encounter

potential people, bump onto

them, see them from afar, but

will never actually get to

know and meet them; even in

the longest time.

The Painful Tangent

Some people are only

meant to meet one

another at one point in

their lives, but are

forever parted.

The Definition of the AsymptoteIt is a straight line associated with a curve

such that as a point moves along infinite branch of the curve the distance from the point to the line approaches zero and the slope of the curve at the point approaches

the slope of the line.

Types of Asymptote

1. Vertical Asymptote2.Horizontal Asymptote3.Oblique Asymptote

Example 53:Find the vertical, horizontal, and

oblique asymptotes (if any) for:

.)4(

1)(

xxf

Type 1: Vertical AsymptoteGiven a rational function:

If f(x) approaches infinity (or

negative infinity) as x approaches

a real number a from the right or

left, then the line x = a is a

VERTICAL ASYMPTOTE.

.0)(;)(

)()( xQ

xQ

xPxf

Theorem 2.1: Vertical Asymptote

If a is a real number such that

Q(a) = 0 and P(a) ≠ 0, then the

line x = a is a vertical asymptote

of the graph f.

01

1

1

01

1

1

...

...

)(

)()(

bxbxbxb

axaxaxa

xQ

xPxf

m

m

m

m

n

n

n

n

Type 2: Horizontal AsymptoteGiven a rational function:

If f(x) approaches infinity (or negative infinity) as f(x)

approaches a real number b, then the line y = b is a HORIZONTAL

ASYMPTOTE.

.0)(;)(

)()( xQ

xQ

xPxf

Theorem 2.2.a: Horizontal Asymptote

The horizontal asymptote of the graph f may be found by the following rules:

If n < m, then y = 0 is a

horizontal asymptote.

01

1

1

01

1

1

...

...

)(

)()(

bxbxbxb

axaxaxa

xQ

xPxf

m

m

m

m

n

n

n

n

Theorem 2.2.b: Horizontal Asymptote


If n=m, thenis a horizontal asymptote.

01

1

1

01

1

1

...

...

)(

)()(

bxbxbxb

axaxaxa

xQ

xPxf

m

m

m

m

n

n

n

n

Theorem 2.2.c: Horizontal Asymptote


If n>m, then there is no horizontal asymptote.

01

1

1

01

1

1

...

...

)(

)()(

bxbxbxb

axaxaxa

xQ

xPxf

m

m

m

m

n

n

n

n

Type 3: Oblique AsymptoteA rational function

has an oblique asymptote if

the degree of P(x) is greater

than the degree of Q(x).

.0)(;)(

)()( xQ

xQ

xPxf

Final Answer: The vertical asymptote of the rational

function is x = 4; The horizontal asymptote is y = 0;

and The rational function does not contain

any oblique asymptote.



.34

423)(

2

2

xx

xxxf


function are x = 1 and x=3; The horizontal asymptote is y = 3;





.9

6)(

2

2

x

xxxf


function is x = -3; The horizontal asymptote is y = 1;





.32

724)(

2

23

xx

xxxf


function are x = -3 and x = 1; The graph has no horizontal

asymptote; and

The oblique asymptote is y = 4x - 6.

Performance Task 8:




independently.

Lecture 10: Graphing Rational Functions



Management (ABM

Mr. Migo M. Mendoza

Example 57:Determine the domain, range,

intercepts, and zeros of the rational function

and sketch the graph.4

1)(

xxf

Take Note:

The technique in graphing rational functions includes

finding the intercepts, zeroes and asymptotes of the rational

function.

Steps in Graphing Rational Function

Step 1:Determine the asymptotes

of the graph.


Step 2:Determine the x-intercepts

and y-intercepts, if there are

any.

Intercepts

The intercepts of the graph of a rational function are the points of intersection of its

graph and an axis.

The Y-Intercept

The y-intecept of the graph of

a rational function r(x), if it

exists, occurs at r(0), provided

that r(x) is defined at x = 0.

The X-InterceptThe x-intercept of the graph of

a rational function r(x), if it exists, occurs at zeroes of the

numerator that are not zeroes of the denominators.


Step 3:Consider the sign of f(x) in the

intervals determined by zeros of

P(x) and Q(x).


Step 4:Identify the symmetry

detected by the test.


Step 5:Plot some points on either side of

each vertical asymptote and check

whether the graph crosses a horizontal

asymptote.


Step 6:Sketch the graph, using the points plotted and using the asymptotes as guide. The graph is a smooth curve, except for breaks at the asymptotes.




6)(

2

2

x

xxxf



of the graph.




any.




P(x) and Q(x).





asymptote.




1)(

2

x

xxf



of the graph.




any.




P(x) and Q(x).





asymptote.

Performance Task 9:




independently.

functionsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter... · 2018-08-31 · lecture 6:...

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