shmth1: general mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · chapter...
TRANSCRIPT
![Page 1: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/1.jpg)
Chapter 1: Functions
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 2: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/2.jpg)
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
![Page 3: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/3.jpg)
Direction:
“The Rule” Game
I will input a photo and using the rule you will give
the output.
![Page 4: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/4.jpg)
“The Rule” Game
THE RULE:
Spit out the author of the book.
![Page 5: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/5.jpg)
My Input:
![Page 6: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/6.jpg)
Your Output:
John Green
![Page 7: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/7.jpg)
“The Rule” Game
THE RULE:
Spit out the title of the movie.
![Page 8: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/8.jpg)
My Input:
![Page 9: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/9.jpg)
Your Output:Fantastic Beasts and Where to
Find Them
![Page 10: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/10.jpg)
“The Rule” Game
THE RULE:
Spit out the title of the TV series.
![Page 11: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/11.jpg)
My Input:
![Page 12: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/12.jpg)
Your Output:
Game of Thrones
![Page 13: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/13.jpg)
“The Rule” Game
THE RULE:
Spit out the name of the famous tourist spot in Rizal.
![Page 14: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/14.jpg)
My Input:
![Page 15: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/15.jpg)
Your Output:
Tinipak River in Tanay, Rizal
![Page 16: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/16.jpg)
“The Rule” Game
THE RULE:
Spit out the name of the Filipino food.
![Page 17: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/17.jpg)
My Input:
![Page 18: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/18.jpg)
Your Output:
Sinigang
![Page 19: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/19.jpg)
“The Rule” Game
THE RULE:
134)( xxf
![Page 20: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/20.jpg)
My Input:
7x
![Page 21: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/21.jpg)
Your Output:
15)7( f
13)7(4)7( f
![Page 22: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/22.jpg)
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?
![Page 23: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/23.jpg)
Lecture 1: Basic Concepts
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 24: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/24.jpg)
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.
![Page 25: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/25.jpg)
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.
![Page 26: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/26.jpg)
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).
![Page 27: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/27.jpg)
![Page 28: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/28.jpg)
A Very Good Explanation of Function:
Function in Real Life:
![Page 29: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/29.jpg)
Did you know?
Functions are often denoted by letter of the
English alphabet or Greek character.
![Page 30: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/30.jpg)
Checking Our Understanding of Function:
Meat-A-Morphosis: An Introduction to
Functions
![Page 31: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/31.jpg)
Domain
It is the set of inputs which serves as entry value to the function
rule.
![Page 32: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/32.jpg)
Codomain
It is the set of outputs.
![Page 33: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/33.jpg)
Range
It is the set of possible outputs.
![Page 34: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/34.jpg)
Something to think about…
What relationship exists between
codomain and range?
![Page 35: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/35.jpg)
Functions as
Transformations and
Mappings
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 36: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/36.jpg)
Did you know?
Functions are often describe as mapping of its
domain unto its range.
![Page 37: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/37.jpg)
The Ellipse/ Arrow Diagram
![Page 38: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/38.jpg)
Understanding Function as Transformation
![Page 39: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/39.jpg)
Classroom Task 2:
Using arrow diagram, which of the following sets of ordered
pairs are functions?
![Page 40: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/40.jpg)
Example 1:
),...}5,4(),4,3(),3,2(),2,1{(M
![Page 41: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/41.jpg)
Final Answer:
Using the Ellipse/ Arrow Diagram, set M is
a function.
![Page 42: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/42.jpg)
Example 2:
)}4,4(),3,3(),2,2(),1,1{(i
![Page 43: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/43.jpg)
Final Answer:
Using the Ellipse/ Arrow Diagram, set i is a
function.
![Page 44: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/44.jpg)
Example 3:
)}1,0(),0,1(),1,0(),0,1{( g
![Page 45: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/45.jpg)
Final Answer:
Using the Ellipse/ Arrow Diagram, set g is
NOT a function.
![Page 46: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/46.jpg)
Example 4:
)}4,2(),1,1(),4,0(),1,1(),4,2{( o
![Page 47: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/47.jpg)
Final Answer:
Using the Ellipse/ Arrow Diagram, set o is
a function.
![Page 48: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/48.jpg)
Example 5:},,,{ zyxwdomM
},,,,,,{ azodnemcodomM
)},(),,(),,(),,{( awmzzyexM
![Page 49: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/49.jpg)
Final Answer:
Using the Ellipse/ Arrow Diagram, set M is
a function.
![Page 50: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/50.jpg)
Example 6:},,,
3
1,
2
1{ dom
},8.0,4.0,1.0{codom
}1.0,,8.0,3
1),4.0,{(
![Page 51: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/51.jpg)
Three Classes of Function
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 52: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/52.jpg)
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.
![Page 53: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/53.jpg)
Something to think about…
Among the examples we have on the board, which
do you think is/are INJECTIVE?
![Page 54: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/54.jpg)
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.
![Page 55: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/55.jpg)
Something to think about…
Among the examples we have on the board, which
do you think is/are SURJECTIVE?
![Page 56: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/56.jpg)
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.
![Page 57: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/57.jpg)
Something to think about…
Among the examples we have on the board, which
do you think is/are BIJECTIVE?
![Page 58: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/58.jpg)
Something to think about…
What relationship should exist between codomain and range before we can say that
a function is onto?
![Page 59: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/59.jpg)
Take Note:
A function is ONTO if RANGE is equal to
CODOMAIN.
![Page 60: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/60.jpg)
Functions in Real Life
Situations
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 61: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/61.jpg)
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?
![Page 62: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/62.jpg)
Final Answer:
Your balance is a function of your bank
account number.
![Page 63: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/63.jpg)
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?
![Page 64: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/64.jpg)
Final Answer:
Apple product is a function of your iPhone
7 Plus.
![Page 65: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/65.jpg)
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?
![Page 66: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/66.jpg)
Final Answer:
Price is a function of the item.
![Page 67: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/67.jpg)
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.
![Page 68: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/68.jpg)
Performance Task 1:
Please download, print
and answer the “Let’s
Practice 1.” Kindly work
independently.
![Page 69: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/69.jpg)
Functions as Formulae,
Rules, and Equations
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 70: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/70.jpg)
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
![Page 71: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/71.jpg)
Final Answer (The Domain):
The domain should be all values of r, r > 0.
![Page 72: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/72.jpg)
Final Answer (The Function Rule):
The function rule is the equation or the formula:
3
3
4rV
![Page 73: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/73.jpg)
Final Answer (The Range):The range should be all values of
volume V that correspond to each
value of radius r.
![Page 74: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/74.jpg)
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.
![Page 75: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/75.jpg)
Final Answer (The Domain):
The domain is all possible values of
distance.
![Page 76: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/76.jpg)
Final Answer (The Function Rule):
The function rule is the equation or the formula:
dreAmountofFa 5.08
![Page 77: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/77.jpg)
Final Answer (The Range):
The range is all amount of fare that corresponds to each amount of distance
d.
![Page 78: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/78.jpg)
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.
![Page 79: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/79.jpg)
Final Answer (The Domain):
The domain is all amount of deposit.
![Page 80: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/80.jpg)
Final Answer (The Function Rule):
The function rule is the equation or formula for computation of interest:
PrI
![Page 81: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/81.jpg)
Final Answer (The Range):
The range is all amount of interest that
corresponds to each amount of deposit.
![Page 82: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/82.jpg)
Graphs of Function
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 83: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/83.jpg)
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.
![Page 84: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/84.jpg)
Classroom Task 3:
Identify which of the given graph represents a function:
![Page 86: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/86.jpg)
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example 8
is a function.
![Page 88: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/88.jpg)
Final Answer:
Therefore, Therefore,
using Vertical Line Test
(VLT) Example 9 is NOT a
function.
![Page 90: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/90.jpg)
Final Answer:
Therefore, Therefore,
using Vertical Line Test
(VLT) Example 10 is a
function.
![Page 92: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/92.jpg)
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example
11 is NOT a function.
![Page 94: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/94.jpg)
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example
12 is a function.
![Page 96: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/96.jpg)
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example
13 is a function.
![Page 97: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/97.jpg)
Understanding Vertical Line Test:
The Vertical Line
Test
![Page 98: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/98.jpg)
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.
![Page 99: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/99.jpg)
Something to think about…• From our previous examples,
can you identify which are injective functions and which
are surjective functions?
![Page 100: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/100.jpg)
Lecture 2: Basic Types of Function
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 101: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/101.jpg)
Type 1: Constant Function
It is a function of zero-degree that are of the form:
where c ≠ 0. ;cxf
![Page 102: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/102.jpg)
For you to Research:
What will happen to the constant function if
c is equal to zero?
![Page 103: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/103.jpg)
Type 2: Linear Function
It is a function of first-degree polynomial that is of
the form:
where c1 ≠ 0. ;01 cxcxf
![Page 104: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/104.jpg)
Something to think about…
What will happen to any linear function if c1
is equal to zero?
![Page 105: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/105.jpg)
Type 3: Quadratic Function
It is a function of the form:
where a ≠ 0. cbxaxxf 2
![Page 106: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/106.jpg)
Something to think about…
What will happen to any quadratic function
if a is equal to zero?
![Page 107: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/107.jpg)
Type 4: Rational Function It is the quotient of two
polynomial functions that are of the form:
where h(x) ≠ 0.
;)(
)(
xh
xgxf
![Page 108: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/108.jpg)
Something to think about…
What will happen to any rational function if
h(x) is equal to zero?
![Page 109: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/109.jpg)
Type 5: Power Function
It is a function of the form:
where n is any real number.
nxxf
![Page 110: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/110.jpg)
Type 6: Absolute Value Function
It is a function of the form:
.xxf
![Page 111: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/111.jpg)
Type 7: Polynomial Function
It is any function f(x) of the form:
.... 01
1
1 cxcxcxcxf n
n
n
n
![Page 112: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/112.jpg)
Something to think about…
What are the conditions that we need to satisfy before we
can say a function is a polynomial?
![Page 113: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/113.jpg)
Not a Polynomial Function:
This is NOT a polynomial function because the variable
has a negative exponent.
521)( xxf
![Page 114: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/114.jpg)
Not a Polynomial Function:
This is NOT a polynomial function because the variable
is in the denominator.
4
3)(
xxf
![Page 115: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/115.jpg)
Not a Polynomial Function:
This is NOT a polynomial function because the variable
is inside a radical.
xxf )(
![Page 116: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/116.jpg)
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
![Page 117: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/117.jpg)
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
![Page 118: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/118.jpg)
Understanding the Greatest Function:
Evaluate the following:
8.
4.1.
7.2.
c
b
a
![Page 119: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/119.jpg)
Final Answer:
Evaluate the following:
88.
24.1.
27.2.
c
b
a
![Page 120: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/120.jpg)
Classroom Task 3:
Determine the type of function of the
following:
![Page 121: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/121.jpg)
Example 16:
Determine the type of function of:
7
3)(
x
xxf
![Page 122: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/122.jpg)
Example 17:
Determine the type of function of:
2483)( 234 xxxxf
![Page 123: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/123.jpg)
Example 18:
Determine the type of function of:
4)( xxf
![Page 124: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/124.jpg)
Example 19:
Determine the type of function of:
18)( xf
![Page 125: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/125.jpg)
Example 20:
Determine the type of function of:
113)( xxf
![Page 126: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/126.jpg)
Example 21:
Determine the type of function of:
12)( xxf
![Page 127: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/127.jpg)
Example 22:
Determine the type of function of:
.65)( 2 xxxf
![Page 128: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/128.jpg)
Example 23:
Determine the type of function of:
.)( xxf
![Page 129: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/129.jpg)
Domain and Range of
Different Functions
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 130: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/130.jpg)
Set of Ordered Pairs:
),...},(),,(),,{( 332211 yxyxyx
![Page 131: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/131.jpg)
Domain and Range of Set of Ordered Pairs:
Domain:
Range:
,...,, 321 xxxdom
,...,, 321 yyyrange
![Page 132: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/132.jpg)
Example 24:
Determine its type, domain and range:
)}.7,6(),5,3(),2,1{(
![Page 133: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/133.jpg)
Something to think about…
If any constant function is of the form: What is
its possible domain and range?
.cxf
![Page 134: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/134.jpg)
Domain and Range of Linear Function:
Domain:
Range:
xxdom |
cyyrange |
![Page 135: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/135.jpg)
Example 25:
Determine its type, domain and range:
.7)( xf
![Page 136: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/136.jpg)
Something to think about…
If any linear function is of the form: What
is its possible domain and range?
.01 cxcxf
![Page 137: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/137.jpg)
Domain and Range of Linear Function:
Domain:
Range:
xxdom |
yyrange |
![Page 138: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/138.jpg)
Example 26:
Determine its type, domain and range:
.92)( xxf
![Page 139: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/139.jpg)
Something to think about…
If any quadratic function is of the form: .
What is its possible domain and range?
cbxaxxf 2
![Page 140: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/140.jpg)
Domain and Range of Quadratic Function:
Condition 1:0;)( 2 acbxaxxf
xxdom |
a
bacyyyrange
4
4,|
2
![Page 141: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/141.jpg)
Example 27:
Determine its type, domain and range:
12)( 2 xxxf
![Page 142: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/142.jpg)
Domain and Range of Quadratic Function:
Condition 2:
0;)( 2 acbxaxxf
xxdom |
a
bacyyyrange
4
4,|
2
![Page 143: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/143.jpg)
Example 28:
Determine its type, domain and range:
.21337)( 2 xxxf
![Page 144: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/144.jpg)
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
![Page 145: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/145.jpg)
Domain and Range of Rational Function:
Domain:
Range:
0)(,| xhxxdom
0)(,| yhyyrange
![Page 146: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/146.jpg)
Example 29:
Determine its type, domain and range:
.2
3)(
x
xxf
![Page 147: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/147.jpg)
How to determine the domain and the range?
Domain:Focus on the denominator.
Range:Solve for x in terms of y.
![Page 148: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/148.jpg)
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 )(
![Page 149: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/149.jpg)
Domain and Range of Radical Function:
Condition 1: n is an odd integerDomain:
Range:
xxdom |
yyrange |
![Page 150: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/150.jpg)
Example 30:
Determine its type, domain and range:
.5)( 3 xxf
![Page 151: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/151.jpg)
Domain and Range of Radical Function:
Condition 1: n is an even integerDomain:
Range:
0)(,| xfxxdom
0,| yyyrange
![Page 152: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/152.jpg)
Example 31:
Determine its type, domain and range:
.8)( xxf
![Page 153: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/153.jpg)
Something to think about…
If any absolute value function is of the form: .
What is its possible domain and range?
xxf
![Page 154: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/154.jpg)
Domain and Range of Absolute Value Function:
Domain:
Range:
xxdom |
0,| yyyrange
![Page 155: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/155.jpg)
Example 32:
Determine its type, domain and range:
.3)( xxf
![Page 156: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/156.jpg)
Performance Task 2:
Please download, print
and answer the “Let’s
Practice 2.” Kindly work
independently.
![Page 157: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/157.jpg)
Lecture 3: Operations on Function
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 158: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/158.jpg)
A Short Recap…
Find the sum of
and3
1
x.
5
2
x
![Page 159: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/159.jpg)
Something to think about…
How do we add two or more
fractions?
![Page 160: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/160.jpg)
Step 1: Adding or Subtracting Fractions
Find the least common denominator (LCD) of both
fractions.
![Page 161: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/161.jpg)
Step 2: Adding or Subtracting Fractions
Rewrite the fractions as equivalent fractions with
the same LCD.
![Page 162: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/162.jpg)
Step 3: Adding or Subtracting Fractions
The LCD is the denominator of the resulting fraction.
![Page 163: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/163.jpg)
Step 3: Adding or Subtracting Fractions
The sum or difference of the numerators is the
numerator of the resulting fraction.
![Page 164: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/164.jpg)
Final Answer:
The sum is:
158
1132
xx
x
![Page 165: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/165.jpg)
A Short Recap…
Find the product of
and23
542
2
xx
xx.
103
652
2
xx
xx
![Page 166: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/166.jpg)
Step 1: Multiplication of Fractions
Rewrite the numerator and denominator in terms
of its prime factors.
![Page 167: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/167.jpg)
Step 2: Multiplication of Fractions
Common factors in the numerator and denominator can be simplified as (this is
often called cancelling).
![Page 168: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/168.jpg)
Step 3: Multiplication of Fractions
Multiply the numerators together to get the new
numerator.
![Page 169: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/169.jpg)
Step 4: Multiplication of Fractions
Multiply the denominators together to get the new
denominator.
![Page 170: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/170.jpg)
Final Answer:
The product is:
2
322
2
xx
xx
![Page 171: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/171.jpg)
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:
![Page 172: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/172.jpg)
Example 33:
)(xgv
Determine the following function:
![Page 173: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/173.jpg)
The Sum of Two or More FunctionsLet f and g be functions:
Their sum, denoted by f + g, is the function defined by
)()()( xgxfxgf
![Page 174: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/174.jpg)
Final Answer:
472)( 2 xxxgv
Thus,
![Page 175: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/175.jpg)
Example 34:
)(xhf
Determine the following function:
![Page 176: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/176.jpg)
Final Answer:
2
13)(
2
x
xxhf
Thus,
![Page 177: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/177.jpg)
Example 35:
)(xpf
Determine the following function:
![Page 178: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/178.jpg)
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
![Page 179: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/179.jpg)
Final Answer:
212)( 2 xxxpf
Thus,
![Page 180: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/180.jpg)
Example 36:
)(xfp
Determine the following function:
![Page 181: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/181.jpg)
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
![Page 182: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/182.jpg)
Final Answer:
10)( xxfp
Thus,
![Page 183: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/183.jpg)
Example 37:
)(xg
v
Determine the following function:
![Page 184: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/184.jpg)
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
![Page 185: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/185.jpg)
Final Answer:
82
45)(
2
2
xx
xxx
g
v
Thus,
![Page 186: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/186.jpg)
Performance Task 4:
Please download, print
and answer the “Let’s
Practice 4.” Kindly work
independently.
![Page 187: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/187.jpg)
A Story of A Famous Burger
![Page 188: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/188.jpg)
Equation for Cooking a Sumptuous Angel’s Burger
Equation for Patty:
Equation for Bread:
1)( xxf
9611)( 2 xxxg
![Page 189: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/189.jpg)
Lecture 4: The Composition of Functions
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 190: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/190.jpg)
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
![Page 191: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/191.jpg)
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.
![Page 192: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/192.jpg)
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.
![Page 193: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/193.jpg)
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
![Page 194: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/194.jpg)
Example 38:
Find and simplify:
)(xfg
![Page 195: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/195.jpg)
Final Answer:
Thus,
22)( xxfg
![Page 196: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/196.jpg)
Example 39:
Find and simplify
Is it the same with
).(xfq
?)(xqf
![Page 197: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/197.jpg)
Final Answer:
The function
and
Thus, they are not the same.
14)( 2 xxfq
.542)( 2 xxxqf
![Page 198: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/198.jpg)
Example 40:
Find and simplify
).(xrf
![Page 199: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/199.jpg)
Final Answer:
Thus,
.1
15)(
x
xxrf
![Page 200: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/200.jpg)
Example 41:
Find and simplify
).5(rF
![Page 201: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/201.jpg)
Final Answer:
Thus,
3)5( rF
![Page 202: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/202.jpg)
Performance Task 5:
Please download, print
and answer the “Let’s
Practice 5.” Kindly work
independently.
![Page 203: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/203.jpg)
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
![Page 204: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/204.jpg)
Lecture 5: The Piecewise Function
SHMth1: General Mathematics
Accountancy, Business and
Management (ABM
Mr. Migo M. Mendoza
![Page 205: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/205.jpg)
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).
![Page 206: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/206.jpg)
Classroom Task 6: What will be the value of:
1
1
)1(
2)(
2 xif
xif
x
xxf
1)(5)(
0)(7)(
xdxb
xcxa
![Page 207: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/207.jpg)
Final Answer:
The value are:
0)1()(16)5()(
2)0()(5)7()(
fdfb
fcfa
![Page 208: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/208.jpg)
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
![Page 209: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/209.jpg)
La Salle College Antipolo
(Cartesian Coordinate Plane)
![Page 210: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/210.jpg)
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
![Page 211: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/211.jpg)
La Salle College Antipolo
(Cartesian Coordinate Plane)
![Page 212: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/212.jpg)
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.
![Page 213: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/213.jpg)
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
![Page 214: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/214.jpg)
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.
![Page 215: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/215.jpg)
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
![Page 216: SHMth1: General Mathematicsmigomendoza.weebly.com/uploads/5/4/7/4/54745209/chapter_1... · Chapter 1: Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM](https://reader031.vdocuments.us/reader031/viewer/2022020100/5a7935f57f8b9ac53b8c19f4/html5/thumbnails/216.jpg)
Performance Task 6:
Please download, print
and answer the “Let’s
Practice 6.” Kindly work
independently.