SlideShare a Scribd company logo
1 of 50
REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
GENERAL MATHEMATICS Samar College Galina V. Panela
What is a function?
Definition of a Function:
• It is a relation define as a set of ordered
pairs (x, y) where no two or more distinct
ordered pairs have the same first element
(x).
• Every value of x corresponds to a unique
value of y
GENERAL MATHEMATICS Samar College Galina V. Panela
Examples:
• Illustrations below are examples of a
function
GENERAL MATHEMATICS Samar College Galina V. Panela
Is it a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
GENERAL MATHEMATICS Samar College Galina V. Panela
What is the difference
between a function and a
relation?
RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS Samar College Galina V. Panela
RELATIONS FUNCTIONS
A relation is a rule that
relates values from a
set of values called the
domain to a second set
of values called the
range.
A function is a relation
where each element in
the domain is related to
only one value in the
range by some rule.
RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS Samar College Galina V. Panela
RELATIONS FUNCTIONS
The elements of the
domain can be imagines
as input to a machine
that applies rule to
these inputs to generate
one or more outputs.
The elements of the
domain can be imagined
as input to a machine
that applies a rule so that
each input corresponds
to only one output.
RELATIONS versus FUNCTIONS
GENERAL MATHEMATICS Samar College Galina V. Panela
RELATIONS FUNCTIONS
A relation is also a set of
ordered pairs (x, y).
A function is a set of
ordered pairs (x, y) such
that no two ordered pairs
have the same x-value
but different y-values.
Is it a function or not?
a. f = {(0, -1), (2, -5), (4, -9), (6,-13)}
b. r ={(a, 0), (b, -1), (c, 0), (d, -1)}
c. g = (5, -10), (25, -50), (50, -100)
d. t = {(-2, 0), (-1, 1), (0, 1), (-2, 2)}
GENERAL MATHEMATICS Samar College Galina V. Panela
The function as a machine…
We will try to represent mathematical
relations as machines with an input
and an output, and that the output is
related to the input by some rule.
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine if this machine produces
a function…
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine if this machine produces
a function…
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine if this machine produces
a function…
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine if this machine produces
a function…
GENERAL MATHEMATICS Samar College Galina V. Panela
GENERAL MATHEMATICS Samar College Galina V. Panela
What is a table of
values?
Table of Values
a
• A table of values is commonly
observed when describing a function.
• This shows the correspondence
between a set of values of x and a set
of values of y in a tabular form.
GENERAL MATHEMATICS Samar College Galina V. Panela
Examples of Table of Values
a
GENERAL MATHEMATICS Samar College Galina V. Panela
x 0 1 4 9 16
y - 5 - 4 - 1 4 11
x -1 -1/4 0 1/4 1
y -1 - 1/2 0 1/2 1
Is it a function or not?
1.A jeepney and its plate number
2.A student and his ID number
3.A teacher and his cellular phone
4.A pen and the color of its ink
GENERAL MATHEMATICS Samar College Galina V. Panela
GENERAL MATHEMATICS Samar College Galina V. Panela
What is a vertical line
test?
Vertical Line Test
a
• The vertical line test for a function
states that if each vertical line
intersects a graph in the x-y plane at
exactly one point, then the graph
illustrates a function.
GENERAL MATHEMATICS Samar College Galina V. Panela
Is this a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
Is this a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
Is this a function or not?
GENERAL MATHEMATICS Samar College Galina V. Panela
Relationship Between the Independent
and Dependent Variables
GENERAL MATHEMATICS Samar College Galina V. Panela
Input
(value of x)
Process
(equation
rule)
Output
(value of y)
Examples:
1. Find the value of y in the equation
y = 10x – 3 if x = - 5.
2. Find the value of x if the value of y
in the equation 𝐲 =
𝟑𝒙+𝟖
𝒙−𝟐
is 2.
GENERAL MATHEMATICS Samar College Galina V. Panela
Applications:
1. A car has travelled a distance of 124
kilometers in 4 hours. Find the speed of the
car.
2. The volume of the cube is defined by the
function 𝑽 = 𝒔 𝟑
where s is the length of the
edge.
• What is the volume of the cube if the
length of the edge is 5 cm?
• What is the length of its edge if its
volume is 728 cubic meters?
GENERAL MATHEMATICS Samar College Galina V. Panela
REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
EVALUATING FUNCTIONS
Evaluating Functions
a
• It is the process of determining the
value of the function at the number
assigned to a given variable.
GENERAL MATHEMATICS Samar College Galina V. Panela
Example:
Let 𝒇(𝒙) = 𝒙 𝟐
− 𝟒𝒙 + 𝟒. Find the following
values of the function
a. f (2)
b. f (-1)
c. f (0)
d. f (- ½ )
e. f (- 4)
GENERAL MATHEMATICS Samar College Galina V. Panela
Example:
Let 𝐠 𝐱 = 𝟑𝒙 − 𝟒. Find the following
values of the function
a. g (2)
b. g (4)
c. g (0)
d. g (9)
e. g (- 1/3)
GENERAL MATHEMATICS Samar College Galina V. Panela
Example:
Let h 𝐱 =
𝟒𝒙+𝟖
𝟐𝒙−𝟒
. Find the following values
of the function
a. h (1)
b. h (-2)
c. h (6)
d. h (0)
e. h (2)
GENERAL MATHEMATICS Samar College Galina V. Panela
REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
DOMAIN AND RANGE OF
FUNCTIONS
Domain D of a Function
a
• It is the set of all x-coordinates in the
set of ordered pairs.
Range R of a Function
a
• It is the set of all y-coordinates in the
set of ordered pairs.
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine the domain and the range
of the following:
a
GENERAL MATHEMATICS Samar College Galina V. Panela
x 0 1 4 9 16
y - 5 - 4 - 1 4 11
x -1 -1/4 0 1/4 1
y -1 - 1/2 0 1/2 1
More on Independent Variables
a
• There are instances in which not all
values of the independent variables
are permissible.
• That is, some functions have
restrictions.
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine the domain and the range
of the following:
a 𝒇 𝒙 =
𝟓
𝒙+𝟐
𝒈 𝒙 = 𝒙 𝟐 − 𝟔𝟒
𝒇 𝒙 =
𝒙 + 𝟑
𝒙 − 𝟐
GENERAL MATHEMATICS Samar College Galina V. Panela
Piece-wise Functions
a
• These are functions which are defined
in defined in different domains since
they are determined by several
equations.
GENERAL MATHEMATICS Samar College Galina V. Panela
Determine the domain and the range
of the following:
a 𝒇 𝒙 =
𝒇 𝒙 =
GENERAL MATHEMATICS Samar College Galina V. Panela
{ 2x + 3 if x ≠ 2
4 if x = 2
{ 2x + 3 if x < 1
– 𝒙 𝟐 + 𝟐 if x ≥ 1
REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
OPERATIONS ON
FUNCTIONS
Operations on Functions
If f and g are functions then
• (f + g) = f(x) + g(x)
• (f – g) = f(x)– g(x)
• (f ∙ g) = f(x) ∙ g(x)
•
𝒇
𝒈
𝒙 =
𝒇(𝒙)
𝒈(𝒙)
where g(x) ≠ 0
GENERAL MATHEMATICS Samar College Galina V. Panela
Example
Let f(x) = 𝒙 𝟐
− 𝟒𝒙 + 𝟑 and g(x)= x – 1.
Perform the operations and identify the
domain
• (f + g)
• (f – g)
• (f ∙ g)
•
𝒇
𝒈
𝒙
GENERAL MATHEMATICS Samar College Galina V. Panela
Example
Let f(x)= x – 3 and g(x) = 𝒙 𝟐
+ 𝟗.
Perform the operations and identify the
domain
• (f + g)
• (f – g)
• (f ∙ g)
•
𝒇
𝒈
𝒙
GENERAL MATHEMATICS Samar College Galina V. Panela
REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
COMPOSITE FUNCTIONS
Operations on Functions
If f and g are functions then the
composite function denoted by 𝒇 ∘ 𝒈, is
defined by
𝒇 ∘ 𝒈 = 𝒇 𝒈(𝒙)
GENERAL MATHEMATICS Samar College Galina V. Panela
Operations on Functions
The domain of 𝒇 ∘ 𝒈 is the set of all
numbers x in the domain of g such that
g(x) is in the domain of f.
GENERAL MATHEMATICS Samar College Galina V. Panela
Example
Let f(x)= x – 3 and g(x) = 𝒙 𝟐
+ 𝟗. Find
• (𝒇 ∘ 𝒈)(x)
• (𝒈 ∘ 𝒇)(x)
• (𝒇 ∘ 𝒈)(3)
• (𝒈 ∘ 𝒇)(- 4)
GENERAL MATHEMATICS Samar College Galina V. Panela
REVIEW ON FUNCTIONS
Module 1
GENERAL MATHEMATICS Samar College Galina V. Panela
EVEN AND ODD
FUNCTIONS
Even and Odd Functions
• A function f is said to be even if
f(–x)=f(x) for each value of x in the
domain of f.
• A function f is said to be odd if
f(–x)= – f(x) for each value of x in the
domain of f.
GENERAL MATHEMATICS Samar College Galina V. Panela
Example
Determine whether each of the following
functions is even, odd or neither
• 𝒇 𝒙 = 𝟒𝒙 𝟒
− 𝟑𝒙 𝟐
− 𝟏𝟎
• 𝒇 𝒙 = −𝒙 𝟓
+ 𝟑𝒙 𝟑
− 𝟏𝟐𝒙
• 𝒇 𝒙 = 𝟒𝒙 𝟑
− 𝟒𝒙 𝟐
− 𝟖𝒙 − 𝟐
GENERAL MATHEMATICS Samar College Galina V. Panela

More Related Content

What's hot

KOmunikasyon at Pananaliksik
KOmunikasyon at PananaliksikKOmunikasyon at Pananaliksik
KOmunikasyon at PananaliksikAlexis Torio
 
GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4
GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4
GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4marvinnbustamante1
 
Heterogeneous at Homogeneous
Heterogeneous at HomogeneousHeterogeneous at Homogeneous
Heterogeneous at HomogeneousThomson Leopoldo
 
Kakayahang Pangkomunikatibo ng mga Pilipino
Kakayahang Pangkomunikatibo ng mga PilipinoKakayahang Pangkomunikatibo ng mga Pilipino
Kakayahang Pangkomunikatibo ng mga PilipinoBasemathBaco
 
Akademikong Sulatin sa Filipino sa Piling Larang
Akademikong Sulatin sa Filipino sa Piling LarangAkademikong Sulatin sa Filipino sa Piling Larang
Akademikong Sulatin sa Filipino sa Piling LarangStemGeneroso
 
Mga Konseptong Pangwika
Mga Konseptong PangwikaMga Konseptong Pangwika
Mga Konseptong PangwikaKaren Fajardo
 
KOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptx
KOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptxKOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptx
KOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptxlucasmonroe1
 
Wikang Opisyal at Wikang Panturo
Wikang Opisyal at Wikang PanturoWikang Opisyal at Wikang Panturo
Wikang Opisyal at Wikang PanturoREGie3
 
Nature and Elements of Communication: Oral Communication in Context
Nature and Elements of Communication: Oral Communication in ContextNature and Elements of Communication: Oral Communication in Context
Nature and Elements of Communication: Oral Communication in Contextmarisolaquino18
 
ORAL COMMUNICATION - TYPES OF SPEECH STYLE
ORAL COMMUNICATION - TYPES OF SPEECH STYLEORAL COMMUNICATION - TYPES OF SPEECH STYLE
ORAL COMMUNICATION - TYPES OF SPEECH STYLEJohn Elmos Seastres
 
Monolingguwalismo, bilingguwalismo, at multilingguwalismo
Monolingguwalismo, bilingguwalismo, at multilingguwalismoMonolingguwalismo, bilingguwalismo, at multilingguwalismo
Monolingguwalismo, bilingguwalismo, at multilingguwalismochxlabastilla
 
KASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILA
KASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILAKASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILA
KASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILAMary Grace Ayade
 
TEKSTONG DESKRIPTIBO - FILIPINO
TEKSTONG DESKRIPTIBO - FILIPINO TEKSTONG DESKRIPTIBO - FILIPINO
TEKSTONG DESKRIPTIBO - FILIPINO NicoleGala
 
HUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATH
HUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATHHUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATH
HUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATHAntonio Delgado
 
Unang wika at Pangalawang wika
Unang wika at Pangalawang wikaUnang wika at Pangalawang wika
Unang wika at Pangalawang wikaAr Jay Bolisay
 

What's hot (20)

Mga Sitwasyong Pangwika sa Pilipinas - GRADE 11
Mga Sitwasyong Pangwika sa Pilipinas - GRADE 11Mga Sitwasyong Pangwika sa Pilipinas - GRADE 11
Mga Sitwasyong Pangwika sa Pilipinas - GRADE 11
 
KOmunikasyon at Pananaliksik
KOmunikasyon at PananaliksikKOmunikasyon at Pananaliksik
KOmunikasyon at Pananaliksik
 
GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4
GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4
GENERAL CHEMISTRY 1 - UNIT 1: Lesson 1-4
 
Heterogeneous at Homogeneous
Heterogeneous at HomogeneousHeterogeneous at Homogeneous
Heterogeneous at Homogeneous
 
Kakayahang Pangkomunikatibo ng mga Pilipino
Kakayahang Pangkomunikatibo ng mga PilipinoKakayahang Pangkomunikatibo ng mga Pilipino
Kakayahang Pangkomunikatibo ng mga Pilipino
 
Akademikong Sulatin sa Filipino sa Piling Larang
Akademikong Sulatin sa Filipino sa Piling LarangAkademikong Sulatin sa Filipino sa Piling Larang
Akademikong Sulatin sa Filipino sa Piling Larang
 
Types of communicative strategies
Types of communicative strategiesTypes of communicative strategies
Types of communicative strategies
 
Mga Konseptong Pangwika
Mga Konseptong PangwikaMga Konseptong Pangwika
Mga Konseptong Pangwika
 
KOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptx
KOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptxKOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptx
KOMUNIKASYON AT PANANALIKSIK SA WIKA AT KULTURANG PILIPINO.pptx
 
Wikang Opisyal at Wikang Panturo
Wikang Opisyal at Wikang PanturoWikang Opisyal at Wikang Panturo
Wikang Opisyal at Wikang Panturo
 
Nature and Elements of Communication: Oral Communication in Context
Nature and Elements of Communication: Oral Communication in ContextNature and Elements of Communication: Oral Communication in Context
Nature and Elements of Communication: Oral Communication in Context
 
ORAL COMMUNICATION - TYPES OF SPEECH STYLE
ORAL COMMUNICATION - TYPES OF SPEECH STYLEORAL COMMUNICATION - TYPES OF SPEECH STYLE
ORAL COMMUNICATION - TYPES OF SPEECH STYLE
 
Lesson 1 Empowerment Technology
Lesson 1 Empowerment TechnologyLesson 1 Empowerment Technology
Lesson 1 Empowerment Technology
 
Monolingguwalismo, bilingguwalismo, at multilingguwalismo
Monolingguwalismo, bilingguwalismo, at multilingguwalismoMonolingguwalismo, bilingguwalismo, at multilingguwalismo
Monolingguwalismo, bilingguwalismo, at multilingguwalismo
 
KASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILA
KASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILAKASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILA
KASAYSAYAN NG WIKANG PAMBANSA- PANAHON NG KASTILA
 
Konseptong Pangwika
Konseptong PangwikaKonseptong Pangwika
Konseptong Pangwika
 
TEKSTONG DESKRIPTIBO - FILIPINO
TEKSTONG DESKRIPTIBO - FILIPINO TEKSTONG DESKRIPTIBO - FILIPINO
TEKSTONG DESKRIPTIBO - FILIPINO
 
Exogenic processes
Exogenic processesExogenic processes
Exogenic processes
 
HUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATH
HUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATHHUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATH
HUMAN PERSONS AS ORIENTED TOWARD THEIR IMPENDING DEATH
 
Unang wika at Pangalawang wika
Unang wika at Pangalawang wikaUnang wika at Pangalawang wika
Unang wika at Pangalawang wika
 

Similar to GENERAL MATHEMATICS Module 1: Review on Functions

Inverse Function.pptx
Inverse Function.pptxInverse Function.pptx
Inverse Function.pptxSerGeo5
 
power point presentation on genmath_lesson1_2_.pptx
power point presentation on genmath_lesson1_2_.pptxpower point presentation on genmath_lesson1_2_.pptx
power point presentation on genmath_lesson1_2_.pptxdatumanongnormalah
 
General Mathematics - Representation and Types of Functions
General Mathematics - Representation and Types of FunctionsGeneral Mathematics - Representation and Types of Functions
General Mathematics - Representation and Types of FunctionsJuan Miguel Palero
 
Powerpoint_Presentation_in_General_Mathe.pdf
Powerpoint_Presentation_in_General_Mathe.pdfPowerpoint_Presentation_in_General_Mathe.pdf
Powerpoint_Presentation_in_General_Mathe.pdfedwardlouieserrano
 
3.1 Functions and Function Notation
3.1 Functions and Function Notation3.1 Functions and Function Notation
3.1 Functions and Function Notationsmiller5
 
Linear Systems - Domain & Range
Linear Systems - Domain & RangeLinear Systems - Domain & Range
Linear Systems - Domain & Rangeswartzje
 
22 01 2014_03_23_31_eee_formula_sheet_final
22 01 2014_03_23_31_eee_formula_sheet_final22 01 2014_03_23_31_eee_formula_sheet_final
22 01 2014_03_23_31_eee_formula_sheet_finalvibhuti bansal
 
PPt on Functions
PPt on FunctionsPPt on Functions
PPt on Functionscoolhanddav
 
Storyboard math
Storyboard mathStoryboard math
Storyboard mathshandex
 
Quadratic Functions.pptx
Quadratic Functions.pptxQuadratic Functions.pptx
Quadratic Functions.pptxMeryAnnMAlday
 
Unit-1 Basic Concept of Algorithm.pptx
Unit-1 Basic Concept of Algorithm.pptxUnit-1 Basic Concept of Algorithm.pptx
Unit-1 Basic Concept of Algorithm.pptxssuser01e301
 
5.1 indentifying linear equations
5.1 indentifying linear equations5.1 indentifying linear equations
5.1 indentifying linear equationscoolhanddav
 
Rational-Function-W3-4.pptx
Rational-Function-W3-4.pptxRational-Function-W3-4.pptx
Rational-Function-W3-4.pptxMYRABACSAFRA2
 

Similar to GENERAL MATHEMATICS Module 1: Review on Functions (20)

Inverse Function.pptx
Inverse Function.pptxInverse Function.pptx
Inverse Function.pptx
 
function
functionfunction
function
 
power point presentation on genmath_lesson1_2_.pptx
power point presentation on genmath_lesson1_2_.pptxpower point presentation on genmath_lesson1_2_.pptx
power point presentation on genmath_lesson1_2_.pptx
 
General Mathematics - Representation and Types of Functions
General Mathematics - Representation and Types of FunctionsGeneral Mathematics - Representation and Types of Functions
General Mathematics - Representation and Types of Functions
 
Powerpoint_Presentation_in_General_Mathe.pdf
Powerpoint_Presentation_in_General_Mathe.pdfPowerpoint_Presentation_in_General_Mathe.pdf
Powerpoint_Presentation_in_General_Mathe.pdf
 
Functions
FunctionsFunctions
Functions
 
3.1 Functions and Function Notation
3.1 Functions and Function Notation3.1 Functions and Function Notation
3.1 Functions and Function Notation
 
Linear Systems - Domain & Range
Linear Systems - Domain & RangeLinear Systems - Domain & Range
Linear Systems - Domain & Range
 
22 01 2014_03_23_31_eee_formula_sheet_final
22 01 2014_03_23_31_eee_formula_sheet_final22 01 2014_03_23_31_eee_formula_sheet_final
22 01 2014_03_23_31_eee_formula_sheet_final
 
One-to-one Functions.pptx
One-to-one Functions.pptxOne-to-one Functions.pptx
One-to-one Functions.pptx
 
Relations and functions
Relations and functionsRelations and functions
Relations and functions
 
PPt on Functions
PPt on FunctionsPPt on Functions
PPt on Functions
 
Functions
FunctionsFunctions
Functions
 
Chapter 5 Identifying Linear Functions
Chapter 5 Identifying Linear FunctionsChapter 5 Identifying Linear Functions
Chapter 5 Identifying Linear Functions
 
Storyboard math
Storyboard mathStoryboard math
Storyboard math
 
Quadratic Functions.pptx
Quadratic Functions.pptxQuadratic Functions.pptx
Quadratic Functions.pptx
 
Unit-1 Basic Concept of Algorithm.pptx
Unit-1 Basic Concept of Algorithm.pptxUnit-1 Basic Concept of Algorithm.pptx
Unit-1 Basic Concept of Algorithm.pptx
 
5.1 indentifying linear equations
5.1 indentifying linear equations5.1 indentifying linear equations
5.1 indentifying linear equations
 
01 FUNCTIONS.pptx
01 FUNCTIONS.pptx01 FUNCTIONS.pptx
01 FUNCTIONS.pptx
 
Rational-Function-W3-4.pptx
Rational-Function-W3-4.pptxRational-Function-W3-4.pptx
Rational-Function-W3-4.pptx
 

Recently uploaded

Using Grammatical Signals Suitable to Patterns of Idea Development
Using Grammatical Signals Suitable to Patterns of Idea DevelopmentUsing Grammatical Signals Suitable to Patterns of Idea Development
Using Grammatical Signals Suitable to Patterns of Idea Developmentchesterberbo7
 
INTRODUCTION TO CATHOLIC CHRISTOLOGY.pptx
INTRODUCTION TO CATHOLIC CHRISTOLOGY.pptxINTRODUCTION TO CATHOLIC CHRISTOLOGY.pptx
INTRODUCTION TO CATHOLIC CHRISTOLOGY.pptxHumphrey A Beña
 
Grade Three -ELLNA-REVIEWER-ENGLISH.pptx
Grade Three -ELLNA-REVIEWER-ENGLISH.pptxGrade Three -ELLNA-REVIEWER-ENGLISH.pptx
Grade Three -ELLNA-REVIEWER-ENGLISH.pptxkarenfajardo43
 
Active Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfActive Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfPatidar M
 
Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationCongestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationdeepaannamalai16
 
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptxDIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptxMichelleTuguinay1
 
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...DhatriParmar
 
4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptx4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptxmary850239
 
How to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 DatabaseHow to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 DatabaseCeline George
 
Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1GloryAnnCastre1
 
How to Fix XML SyntaxError in Odoo the 17
How to Fix XML SyntaxError in Odoo the 17How to Fix XML SyntaxError in Odoo the 17
How to Fix XML SyntaxError in Odoo the 17Celine George
 
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITWQ-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITWQuiz Club NITW
 
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITWQ-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITWQuiz Club NITW
 
Textual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHSTextual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHSMae Pangan
 
Narcotic and Non Narcotic Analgesic..pdf
Narcotic and Non Narcotic Analgesic..pdfNarcotic and Non Narcotic Analgesic..pdf
Narcotic and Non Narcotic Analgesic..pdfPrerana Jadhav
 
MS4 level being good citizen -imperative- (1) (1).pdf
MS4 level   being good citizen -imperative- (1) (1).pdfMS4 level   being good citizen -imperative- (1) (1).pdf
MS4 level being good citizen -imperative- (1) (1).pdfMr Bounab Samir
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfVanessa Camilleri
 
4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptxmary850239
 

Recently uploaded (20)

Using Grammatical Signals Suitable to Patterns of Idea Development
Using Grammatical Signals Suitable to Patterns of Idea DevelopmentUsing Grammatical Signals Suitable to Patterns of Idea Development
Using Grammatical Signals Suitable to Patterns of Idea Development
 
INTRODUCTION TO CATHOLIC CHRISTOLOGY.pptx
INTRODUCTION TO CATHOLIC CHRISTOLOGY.pptxINTRODUCTION TO CATHOLIC CHRISTOLOGY.pptx
INTRODUCTION TO CATHOLIC CHRISTOLOGY.pptx
 
Grade Three -ELLNA-REVIEWER-ENGLISH.pptx
Grade Three -ELLNA-REVIEWER-ENGLISH.pptxGrade Three -ELLNA-REVIEWER-ENGLISH.pptx
Grade Three -ELLNA-REVIEWER-ENGLISH.pptx
 
Active Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfActive Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdf
 
Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationCongestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentation
 
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptxDIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
 
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
 
4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptx4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptx
 
How to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 DatabaseHow to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 Database
 
Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1
 
INCLUSIVE EDUCATION PRACTICES FOR TEACHERS AND TRAINERS.pptx
INCLUSIVE EDUCATION PRACTICES FOR TEACHERS AND TRAINERS.pptxINCLUSIVE EDUCATION PRACTICES FOR TEACHERS AND TRAINERS.pptx
INCLUSIVE EDUCATION PRACTICES FOR TEACHERS AND TRAINERS.pptx
 
How to Fix XML SyntaxError in Odoo the 17
How to Fix XML SyntaxError in Odoo the 17How to Fix XML SyntaxError in Odoo the 17
How to Fix XML SyntaxError in Odoo the 17
 
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITWQ-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
 
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITWQ-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
 
prashanth updated resume 2024 for Teaching Profession
prashanth updated resume 2024 for Teaching Professionprashanth updated resume 2024 for Teaching Profession
prashanth updated resume 2024 for Teaching Profession
 
Textual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHSTextual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHS
 
Narcotic and Non Narcotic Analgesic..pdf
Narcotic and Non Narcotic Analgesic..pdfNarcotic and Non Narcotic Analgesic..pdf
Narcotic and Non Narcotic Analgesic..pdf
 
MS4 level being good citizen -imperative- (1) (1).pdf
MS4 level   being good citizen -imperative- (1) (1).pdfMS4 level   being good citizen -imperative- (1) (1).pdf
MS4 level being good citizen -imperative- (1) (1).pdf
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdf
 
4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx
 

GENERAL MATHEMATICS Module 1: Review on Functions

  • 1. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela
  • 2. GENERAL MATHEMATICS Samar College Galina V. Panela What is a function?
  • 3. Definition of a Function: • It is a relation define as a set of ordered pairs (x, y) where no two or more distinct ordered pairs have the same first element (x). • Every value of x corresponds to a unique value of y GENERAL MATHEMATICS Samar College Galina V. Panela
  • 4. Examples: • Illustrations below are examples of a function GENERAL MATHEMATICS Samar College Galina V. Panela
  • 5. Is it a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela
  • 6. GENERAL MATHEMATICS Samar College Galina V. Panela What is the difference between a function and a relation?
  • 7. RELATIONS versus FUNCTIONS GENERAL MATHEMATICS Samar College Galina V. Panela RELATIONS FUNCTIONS A relation is a rule that relates values from a set of values called the domain to a second set of values called the range. A function is a relation where each element in the domain is related to only one value in the range by some rule.
  • 8. RELATIONS versus FUNCTIONS GENERAL MATHEMATICS Samar College Galina V. Panela RELATIONS FUNCTIONS The elements of the domain can be imagines as input to a machine that applies rule to these inputs to generate one or more outputs. The elements of the domain can be imagined as input to a machine that applies a rule so that each input corresponds to only one output.
  • 9. RELATIONS versus FUNCTIONS GENERAL MATHEMATICS Samar College Galina V. Panela RELATIONS FUNCTIONS A relation is also a set of ordered pairs (x, y). A function is a set of ordered pairs (x, y) such that no two ordered pairs have the same x-value but different y-values.
  • 10. Is it a function or not? a. f = {(0, -1), (2, -5), (4, -9), (6,-13)} b. r ={(a, 0), (b, -1), (c, 0), (d, -1)} c. g = (5, -10), (25, -50), (50, -100) d. t = {(-2, 0), (-1, 1), (0, 1), (-2, 2)} GENERAL MATHEMATICS Samar College Galina V. Panela
  • 11. The function as a machine… We will try to represent mathematical relations as machines with an input and an output, and that the output is related to the input by some rule. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 12. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela
  • 13. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela
  • 14. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela
  • 15. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela
  • 16. GENERAL MATHEMATICS Samar College Galina V. Panela What is a table of values?
  • 17. Table of Values a • A table of values is commonly observed when describing a function. • This shows the correspondence between a set of values of x and a set of values of y in a tabular form. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 18. Examples of Table of Values a GENERAL MATHEMATICS Samar College Galina V. Panela x 0 1 4 9 16 y - 5 - 4 - 1 4 11 x -1 -1/4 0 1/4 1 y -1 - 1/2 0 1/2 1
  • 19. Is it a function or not? 1.A jeepney and its plate number 2.A student and his ID number 3.A teacher and his cellular phone 4.A pen and the color of its ink GENERAL MATHEMATICS Samar College Galina V. Panela
  • 20. GENERAL MATHEMATICS Samar College Galina V. Panela What is a vertical line test?
  • 21. Vertical Line Test a • The vertical line test for a function states that if each vertical line intersects a graph in the x-y plane at exactly one point, then the graph illustrates a function. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 22. Is this a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela
  • 23. Is this a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela
  • 24. Is this a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela
  • 25. Relationship Between the Independent and Dependent Variables GENERAL MATHEMATICS Samar College Galina V. Panela Input (value of x) Process (equation rule) Output (value of y)
  • 26. Examples: 1. Find the value of y in the equation y = 10x – 3 if x = - 5. 2. Find the value of x if the value of y in the equation 𝐲 = 𝟑𝒙+𝟖 𝒙−𝟐 is 2. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 27. Applications: 1. A car has travelled a distance of 124 kilometers in 4 hours. Find the speed of the car. 2. The volume of the cube is defined by the function 𝑽 = 𝒔 𝟑 where s is the length of the edge. • What is the volume of the cube if the length of the edge is 5 cm? • What is the length of its edge if its volume is 728 cubic meters? GENERAL MATHEMATICS Samar College Galina V. Panela
  • 28. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela EVALUATING FUNCTIONS
  • 29. Evaluating Functions a • It is the process of determining the value of the function at the number assigned to a given variable. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 30. Example: Let 𝒇(𝒙) = 𝒙 𝟐 − 𝟒𝒙 + 𝟒. Find the following values of the function a. f (2) b. f (-1) c. f (0) d. f (- ½ ) e. f (- 4) GENERAL MATHEMATICS Samar College Galina V. Panela
  • 31. Example: Let 𝐠 𝐱 = 𝟑𝒙 − 𝟒. Find the following values of the function a. g (2) b. g (4) c. g (0) d. g (9) e. g (- 1/3) GENERAL MATHEMATICS Samar College Galina V. Panela
  • 32. Example: Let h 𝐱 = 𝟒𝒙+𝟖 𝟐𝒙−𝟒 . Find the following values of the function a. h (1) b. h (-2) c. h (6) d. h (0) e. h (2) GENERAL MATHEMATICS Samar College Galina V. Panela
  • 33. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela DOMAIN AND RANGE OF FUNCTIONS
  • 34. Domain D of a Function a • It is the set of all x-coordinates in the set of ordered pairs. Range R of a Function a • It is the set of all y-coordinates in the set of ordered pairs. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 35. Determine the domain and the range of the following: a GENERAL MATHEMATICS Samar College Galina V. Panela x 0 1 4 9 16 y - 5 - 4 - 1 4 11 x -1 -1/4 0 1/4 1 y -1 - 1/2 0 1/2 1
  • 36. More on Independent Variables a • There are instances in which not all values of the independent variables are permissible. • That is, some functions have restrictions. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 37. Determine the domain and the range of the following: a 𝒇 𝒙 = 𝟓 𝒙+𝟐 𝒈 𝒙 = 𝒙 𝟐 − 𝟔𝟒 𝒇 𝒙 = 𝒙 + 𝟑 𝒙 − 𝟐 GENERAL MATHEMATICS Samar College Galina V. Panela
  • 38. Piece-wise Functions a • These are functions which are defined in defined in different domains since they are determined by several equations. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 39. Determine the domain and the range of the following: a 𝒇 𝒙 = 𝒇 𝒙 = GENERAL MATHEMATICS Samar College Galina V. Panela { 2x + 3 if x ≠ 2 4 if x = 2 { 2x + 3 if x < 1 – 𝒙 𝟐 + 𝟐 if x ≥ 1
  • 40. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela OPERATIONS ON FUNCTIONS
  • 41. Operations on Functions If f and g are functions then • (f + g) = f(x) + g(x) • (f – g) = f(x)– g(x) • (f ∙ g) = f(x) ∙ g(x) • 𝒇 𝒈 𝒙 = 𝒇(𝒙) 𝒈(𝒙) where g(x) ≠ 0 GENERAL MATHEMATICS Samar College Galina V. Panela
  • 42. Example Let f(x) = 𝒙 𝟐 − 𝟒𝒙 + 𝟑 and g(x)= x – 1. Perform the operations and identify the domain • (f + g) • (f – g) • (f ∙ g) • 𝒇 𝒈 𝒙 GENERAL MATHEMATICS Samar College Galina V. Panela
  • 43. Example Let f(x)= x – 3 and g(x) = 𝒙 𝟐 + 𝟗. Perform the operations and identify the domain • (f + g) • (f – g) • (f ∙ g) • 𝒇 𝒈 𝒙 GENERAL MATHEMATICS Samar College Galina V. Panela
  • 44. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela COMPOSITE FUNCTIONS
  • 45. Operations on Functions If f and g are functions then the composite function denoted by 𝒇 ∘ 𝒈, is defined by 𝒇 ∘ 𝒈 = 𝒇 𝒈(𝒙) GENERAL MATHEMATICS Samar College Galina V. Panela
  • 46. Operations on Functions The domain of 𝒇 ∘ 𝒈 is the set of all numbers x in the domain of g such that g(x) is in the domain of f. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 47. Example Let f(x)= x – 3 and g(x) = 𝒙 𝟐 + 𝟗. Find • (𝒇 ∘ 𝒈)(x) • (𝒈 ∘ 𝒇)(x) • (𝒇 ∘ 𝒈)(3) • (𝒈 ∘ 𝒇)(- 4) GENERAL MATHEMATICS Samar College Galina V. Panela
  • 48. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela EVEN AND ODD FUNCTIONS
  • 49. Even and Odd Functions • A function f is said to be even if f(–x)=f(x) for each value of x in the domain of f. • A function f is said to be odd if f(–x)= – f(x) for each value of x in the domain of f. GENERAL MATHEMATICS Samar College Galina V. Panela
  • 50. Example Determine whether each of the following functions is even, odd or neither • 𝒇 𝒙 = 𝟒𝒙 𝟒 − 𝟑𝒙 𝟐 − 𝟏𝟎 • 𝒇 𝒙 = −𝒙 𝟓 + 𝟑𝒙 𝟑 − 𝟏𝟐𝒙 • 𝒇 𝒙 = 𝟒𝒙 𝟑 − 𝟒𝒙 𝟐 − 𝟖𝒙 − 𝟐 GENERAL MATHEMATICS Samar College Galina V. Panela