Chapter 1: Functions

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Chapter 1: Functions Lecture 1: Basic Concepts

Lecture 2: Basic Types of Functions

Lecture 3: Operations on Function

Lecture 4: The Composition of

Functions Lecture 5: The Piecewise Function

Direction:

“The Rule” Game

I will input a photo and using the rule you will give

the output.

“The Rule” Game

THE RULE:

Spit out the author of the book.

My Input:

Your Output:

John Green

“The Rule” Game

THE RULE:

Spit out the title of the movie.

My Input:

Your Output:Fantastic Beasts and Where to

Find Them

“The Rule” Game

THE RULE:

Spit out the title of the TV series.

My Input:

Your Output:

Game of Thrones

“The Rule” Game

THE RULE:

Spit out the name of the famous tourist spot in Rizal.

My Input:

Your Output:

Tinipak River in Tanay, Rizal

“The Rule” Game

THE RULE:

Spit out the name of the Filipino food.

My Input:

Your Output:

Sinigang

“The Rule” Game

THE RULE:

134)( xxf

My Input:

7x

Your Output:

15)7( f

13)7(4)7( f

Activity Processing:

From our activity, is it possible that for every input (that I gave),

you can give two or more possible output? Why or why

not?

Lecture 1: Basic Concepts

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Function It is a rule of a relationship

between an input (independent) quantity and an output (dependent) quantity in which each input value

uniquely determine one output value.

The Formal Definition of FunctionLet A and B be sets. A function

from A to B, denoted ƒ : A → B, is a relation from A to B such that, for

every a ∈ A, there exists a unique b ∈B such that (a, b) ∈ ƒ.

Take Note:If ƒ is a function and x is an

element in its domain, then, to each element x, the function ƒ associates exactly one value to

be denoted by ƒ(x).

A Very Good Explanation of Function:

Function in Real Life:

Did you know?

Functions are often denoted by letter of the

English alphabet or Greek character.

Checking Our Understanding of Function:

Meat-A-Morphosis: An Introduction to

Functions

Domain

It is the set of inputs which serves as entry value to the function

rule.

Codomain

It is the set of outputs.

Range

It is the set of possible outputs.

Something to think about…

What relationship exists between

codomain and range?

Functions as

Transformations and

Mappings

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Did you know?

Functions are often describe as mapping of its

domain unto its range.

The Ellipse/ Arrow Diagram

Understanding Function as Transformation

Classroom Task 2:

Using arrow diagram, which of the following sets of ordered

pairs are functions?

Example 1:

),...}5,4(),4,3(),3,2(),2,1{(M

Final Answer:

Using the Ellipse/ Arrow Diagram, set M is

a function.

Example 2:

)}4,4(),3,3(),2,2(),1,1{(i

Final Answer:

Using the Ellipse/ Arrow Diagram, set i is a

function.

Example 3:

)}1,0(),0,1(),1,0(),0,1{( g

Final Answer:

Using the Ellipse/ Arrow Diagram, set g is

NOT a function.

Example 4:

)}4,2(),1,1(),4,0(),1,1(),4,2{( o

Final Answer:

Using the Ellipse/ Arrow Diagram, set o is

a function.

Example 5:},,,{ zyxwdomM

},,,,,,{ azodnemcodomM

)},(),,(),,(),,{( awmzzyexM

Final Answer:

Using the Ellipse/ Arrow Diagram, set M is

a function.

Example 6:},,,

3

1,

2

1{ dom

},8.0,4.0,1.0{codom

}1.0,,8.0,3

1),4.0,{(

Three Classes of Function

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Injective Function (One-to-One Function)

A function ƒ : A → B is said to be injective (or one-to-one) if for each b ∈ B, there is at most one a

∈ A for which ƒ (a) = b.

Something to think about…

Among the examples we have on the board, which

do you think is/are INJECTIVE?

Surjective Function (Onto)A function ƒ : A → B is said to be surjective (or onto) if for eachb ∈ B, there exist a ∈ A for which

ƒ (a) = b.

Something to think about…

Among the examples we have on the board, which

do you think is/are SURJECTIVE?

Bijective Function (One-to-One Correspondence)

A function that is both injective and surjective is said to be bijective or a one-to-one

correspondence.

Something to think about…

Among the examples we have on the board, which

do you think is/are BIJECTIVE?

Something to think about…

What relationship should exist between codomain and range before we can say that

a function is onto?

Take Note:

A function is ONTO if RANGE is equal to

CODOMAIN.

Functions in Real Life

Situations

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Real Life Situation # 1:Let's consider bank account information…

Is your balance a function of your back account? Is your

bank account number a function of your balance?

Final Answer:

Your balance is a function of your bank

account number.

Real Life Situation # 2: In choosing a smartphone. Is your iPhone 7 Plus a

function of Apple product? Or is Apple product a

function of your iPhone 7 Plus?

Final Answer:

Apple product is a function of your iPhone

7 Plus.

Real Life Situation # 3:At a coffee shop, the menu consists of

items and their prices. Is price a function of the item? Is the item a function of the

price?

Final Answer:

Price is a function of the item.

Classroom Task 1:

Give your own example of a function you usually experience

everyday. Share to us why do you think it is a function.

Performance Task 1:

Please download, print

and answer the “Let’s

Practice 1.” Kindly work

independently.

Functions as Formulae,

Rules, and Equations

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Example 7: Identify the domain, the function rule, and the

range of the following:

The volume of spherical balloon of radius r is given by:

3

3

4rV

Final Answer (The Domain):

The domain should be all values of r, r > 0.

Final Answer (The Function Rule):

The function rule is the equation or the formula:

3

3

4rV

Final Answer (The Range):The range should be all values of

volume V that correspond to each

value of radius r.

Example 8: Identify the domain, the function rule, and

the range of the following:

A jeepney passenger pays Php 8.00 for the first 5 km as fare, and an

additional Php0.50 for every succeeding distance d in kilometer.

Final Answer (The Domain):

The domain is all possible values of

distance.

Final Answer (The Function Rule):

The function rule is the equation or the formula:

dreAmountofFa 5.08

Final Answer (The Range):

The range is all amount of fare that corresponds to each amount of distance

d.

Example 9: Identify the domain, the function rule, and

the range of the following:

The interest earned by a bank investment depends on the

amount of deposit.

Final Answer (The Domain):

The domain is all amount of deposit.

Final Answer (The Function Rule):

The function rule is the equation or formula for computation of interest:

PrI

Final Answer (The Range):

The range is all amount of interest that

corresponds to each amount of deposit.

Graphs of Function

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

The Vertical Line Test (VLT)A graph of a mathematical relation

is a function if any vertical line drawn passing through the graph intersects the graph at exactly one

point.

Classroom Task 3:

Identify which of the given graph represents a function:

Example 10:

Final Answer:

Therefore, using Vertical

Line Test (VLT) Example 8

is a function.

Example 11:

Final Answer:

Therefore, Therefore,

using Vertical Line Test

(VLT) Example 9 is NOT a

function.

Example 12:

Final Answer:

Therefore, Therefore,

using Vertical Line Test

(VLT) Example 10 is a

function.

Example 13:

Final Answer:

Therefore, using Vertical

Line Test (VLT) Example

11 is NOT a function.

Example 14:

Final Answer:

Therefore, using Vertical

Line Test (VLT) Example

12 is a function.

Example 15:

Final Answer:

Therefore, using Vertical

Line Test (VLT) Example

13 is a function.

Understanding Vertical Line Test:

The Vertical Line

Test

The Horizontal Line Test (HLT)• The horizontal line test is a test used

to determine whether a function is injective (i.e., one-to-one). If any

horizontal line y = c intersects the

graph in more than one point, the function is not injective.

Something to think about…• From our previous examples,

can you identify which are injective functions and which

are surjective functions?

Lecture 2: Basic Types of Function

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Type 1: Constant Function

It is a function of zero-degree that are of the form:

where c ≠ 0. ;cxf

For you to Research:

What will happen to the constant function if

c is equal to zero?

Type 2: Linear Function

It is a function of first-degree polynomial that is of

the form:

where c1 ≠ 0. ;01 cxcxf

Something to think about…

What will happen to any linear function if c1

is equal to zero?

Type 3: Quadratic Function

It is a function of the form:

where a ≠ 0. cbxaxxf 2

Something to think about…

What will happen to any quadratic function

if a is equal to zero?

Type 4: Rational Function It is the quotient of two

polynomial functions that are of the form:

where h(x) ≠ 0.

;)(

)(

xh

xgxf

Something to think about…

What will happen to any rational function if

h(x) is equal to zero?

Type 5: Power Function

It is a function of the form:

where n is any real number.

nxxf

Type 6: Absolute Value Function

It is a function of the form:

.xxf

Type 7: Polynomial Function

It is any function f(x) of the form:

.... 01

1

1 cxcxcxcxf n

n

n

n

Something to think about…

What are the conditions that we need to satisfy before we

can say a function is a polynomial?

Not a Polynomial Function:

This is NOT a polynomial function because the variable

has a negative exponent.

521)( xxf

Not a Polynomial Function:

This is NOT a polynomial function because the variable

is in the denominator.

4

3)(

xxf

Not a Polynomial Function:

This is NOT a polynomial function because the variable

is inside a radical.

xxf )(

Type 8: The Greatest Function

It is any function f(x) of the form: The value is the greatest integer that is less

than or equal to x.

.xxf x

The Greatest Integer or Floor Function

The symbol denotes the greatest integer or floor function applied to d.

The floor function gives the largest integer less than or equal to d.

Example:

d

49.41.4

Understanding the Greatest Function:

Evaluate the following:

8.

4.1.

7.2.

c

b

a

Final Answer:

Evaluate the following:

88.

24.1.

27.2.

c

b

a

Classroom Task 3:

Determine the type of function of the

following:

Example 16:

Determine the type of function of:

7

3)(

x

xxf

Example 17:

Determine the type of function of:

2483)( 234 xxxxf

Example 18:

Determine the type of function of:

4)( xxf

Example 19:

Determine the type of function of:

18)( xf

Example 20:

Determine the type of function of:

113)( xxf

Example 21:

Determine the type of function of:

12)( xxf

Example 22:

Determine the type of function of:

.65)( 2 xxxf

Example 23:

Determine the type of function of:

.)( xxf

Domain and Range of

Different Functions

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Set of Ordered Pairs:

),...},(),,(),,{( 332211 yxyxyx

Domain and Range of Set of Ordered Pairs:

Domain:

Range:

,...,, 321 xxxdom

,...,, 321 yyyrange

Example 24:

Determine its type, domain and range:

)}.7,6(),5,3(),2,1{(

Something to think about…

If any constant function is of the form: What is

its possible domain and range?

.cxf

Domain and Range of Linear Function:

Domain:

Range:

xxdom |

cyyrange |

Example 25:

Determine its type, domain and range:

.7)( xf

Something to think about…

If any linear function is of the form: What

is its possible domain and range?

.01 cxcxf

Domain and Range of Linear Function:

Domain:

Range:

xxdom |

yyrange |

Example 26:

Determine its type, domain and range:

.92)( xxf

Something to think about…

If any quadratic function is of the form: .

What is its possible domain and range?

cbxaxxf 2

Domain and Range of Quadratic Function:

Condition 1:0;)( 2 acbxaxxf

xxdom |

a

bacyyyrange

4

4,|

2

Example 27:

Determine its type, domain and range:

12)( 2 xxxf

Domain and Range of Quadratic Function:

Condition 2:

0;)( 2 acbxaxxf

xxdom |

a

bacyyyrange

4

4,|

2

Example 28:

Determine its type, domain and range:

.21337)( 2 xxxf

Something to think about…

If any rational function is of the form: where

h(x) ≠ 0. What is its possible domain and range?

;)(

)(

xh

xgxf

Domain and Range of Rational Function:

Domain:

Range:

0)(,| xhxxdom

0)(,| yhyyrange

Example 29:

Determine its type, domain and range:

.2

3)(

x

xxf

How to determine the domain and the range?

Domain:Focus on the denominator.

Range:Solve for x in terms of y.

Something to think about…What are the possible domain and

range of RADICAL FUNCTION:

n is an odd integer,n is an even integer,

n xfy )(

Domain and Range of Radical Function:

Condition 1: n is an odd integerDomain:

Range:

xxdom |

yyrange |

Example 30:

Determine its type, domain and range:

.5)( 3 xxf

Domain and Range of Radical Function:

Condition 1: n is an even integerDomain:

Range:

0)(,| xfxxdom

0,| yyyrange

Example 31:

Determine its type, domain and range:

.8)( xxf

Something to think about…

If any absolute value function is of the form: .

What is its possible domain and range?

xxf

Domain and Range of Absolute Value Function:

Domain:

Range:

xxdom |

0,| yyyrange

Example 32:

Determine its type, domain and range:

.3)( xxf

Performance Task 2:

Please download, print

and answer the “Let’s

Practice 2.” Kindly work

independently.

Lecture 3: Operations on Function

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

A Short Recap…

Find the sum of

and3

1

x.

5

2

x

Something to think about…

How do we add two or more

fractions?

Step 1: Adding or Subtracting Fractions

Find the least common denominator (LCD) of both

fractions.

Step 2: Adding or Subtracting Fractions

Rewrite the fractions as equivalent fractions with

the same LCD.

Step 3: Adding or Subtracting Fractions

The LCD is the denominator of the resulting fraction.

Step 3: Adding or Subtracting Fractions

The sum or difference of the numerators is the

numerator of the resulting fraction.

Final Answer:

The sum is:

158

1132

xx

x

A Short Recap…

Find the product of

and23

542

2

xx

xx.

103

652

2

xx

xx

Step 1: Multiplication of Fractions

Rewrite the numerator and denominator in terms

of its prime factors.

Step 2: Multiplication of Fractions

Common factors in the numerator and denominator can be simplified as (this is

often called cancelling).

Step 3: Multiplication of Fractions

Multiply the numerators together to get the new

numerator.

Step 4: Multiplication of Fractions

Multiply the denominators together to get the new

denominator.

Final Answer:

The product is:

2

322

2

xx

xx

Classroom Task 4:

3)( xxf72)( xxp

45)( 2 xxxv

82)( 2 xxxg

x

xxh

2

7)(

3

2)(

x

xxt

Given the functions below, determine the following functions on next slides:

Example 33:

)(xgv

Determine the following function:

The Sum of Two or More FunctionsLet f and g be functions:

Their sum, denoted by f + g, is the function defined by

)()()( xgxfxgf

Final Answer:

472)( 2 xxxgv

Thus,

Example 34:

)(xhf

Determine the following function:

Final Answer:

2

13)(

2

x

xxhf

Thus,

Example 35:

)(xpf

Determine the following function:

The Product of Two or More Functions

Let f and g be functions:

Their product, denoted by f ⦁ g, is the function defined by

)()()( xgxfxgf

Final Answer:

212)( 2 xxxpf

Thus,

Example 36:

)(xfp

Determine the following function:

The Difference of Two or More Functions

Let f and g be functions:

Their difference, denoted by f - g, is the function defined by

)()()( xgxfxgf

Final Answer:

10)( xxfp

Thus,

Example 37:

)(xg

v

Determine the following function:

The Quotient of Two or More Functions

Let f and g be functions:

Their quotient, denoted by f ⦁ g, is the function defined by

.)(

)()(

xg

xfx

g

f

Final Answer:

82

45)(

2

2

xx

xxx

g

v

Thus,

Performance Task 4:

Please download, print

and answer the “Let’s

Practice 4.” Kindly work

independently.

A Story of A Famous Burger

Equation for Cooking a Sumptuous Angel’s Burger

Equation for Patty:

Equation for Bread:

1)( xxf

9611)( 2 xxxg

Lecture 4: The Composition of Functions

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Formal Definition of Composition of Function

Let f and g be functions. The composite function, denoted by (f ○ g), is

defined by:

The process of obtaining a composite function is called

function composition.

)()(g xgfxf

Composition of Function It is another way of combining

functions. This method of combining functions uses the output of one

function as the input for the second function.

The Domain of the Composite Function

The domain of (f ○ g) is the set of all numbers x in the domain of g that g(x) is in

the domain of f.

Classroom Task 5: Given the following functions, find and simplify

the given composite functions on the next slides:

12)( xxf

22)( 2 xxxq

1

12)(

x

xxr

1)( xxg

1)( xxF

Example 38:

Find and simplify:

)(xfg

Final Answer:

Thus,

22)( xxfg

Example 39:

Find and simplify

Is it the same with

).(xfq

?)(xqf

Final Answer:

The function

and

Thus, they are not the same.

14)( 2 xxfq

.542)( 2 xxxqf

Example 40:

Find and simplify

).(xrf

Final Answer:

Thus,

.1

15)(

x

xxrf

Example 41:

Find and simplify

).5(rF

Final Answer:

Thus,

3)5( rF

Performance Task 5:

Please download, print

and answer the “Let’s

Practice 5.” Kindly work

independently.

Something to think about… Have you encountered this type of function

before? What does this type of equation tell us?

1

1

)1(

2)(

2 xif

xif

x

xxf

Lecture 5: The Piecewise Function

SHMth1: General Mathematics

Accountancy, Business and

Management (ABM

Mr. Migo M. Mendoza

Piecewise-Defined FunctionAlso called as Hybrid Function, is a function which is defined by multiple

sub-functions where each sub-function applied to a certain interval of the main

functions domain (a sub-domain).

Classroom Task 6: What will be the value of:

1

1

)1(

2)(

2 xif

xif

x

xxf

1)(5)(

0)(7)(

xdxb

xcxa

Final Answer:

The value are:

0)1()(16)5()(

2)0()(5)7()(

fdfb

fcfa

Example 42:Given the function:

Find f(x) at x = -6, -3, -1, 0, 2, 3 and 4. Sketch the graph of the piecewise function.

1

1

)1(

2)(

2 xif

xif

x

xxf

La Salle College Antipolo

(Cartesian Coordinate Plane)

Example 43:Given the function:

Find f(x) at x = 0, 1, 2, 3 and 4. Sketch the graph of the piecewise function.

33

312

11

)(

xif

xif

xif

xf

La Salle College Antipolo

(Cartesian Coordinate Plane)

Example 44: Madam Lily Mangipin is charged P300.00 monthly for a particular mobile plan, which includes 100 free

text messages. Messages in excess of 100 are charged P1.00 each. Represent the amount a

consumer pays each month as a function of the number of messages in excess m sent in a month.

Final Answer: The amount a consumer pays each month as a function of the number of messages m

sent in a month is:

100

1000

300

300)(

ifm

mif

mmt

Example 45: A jeepney ride costs P8.00 for the first

4 kilometers, and each additional integer kilometer adds P1.50 to the fare. Use a piecewise function to represent the jeepney fare in terms of each additional

distance d in kilometers.

Final Answer: The piecewise function that represents the

jeepney fare in terms of the distance d in kilometers is:

4

40

5.18

8)(

ifd

dif

ddF

Performance Task 6:

Please download, print

and answer the “Let’s

Practice 6.” Kindly work

independently.