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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:
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example 8
is a function.
Final Answer:
Therefore, Therefore,
using Vertical Line Test
(VLT) Example 9 is NOT a
function.
Final Answer:
Therefore, Therefore,
using Vertical Line Test
(VLT) Example 10 is a
function.
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example
11 is NOT a function.
Final Answer:
Therefore, using Vertical
Line Test (VLT) Example
12 is a function.
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.