General Mathematics – Grade 11

The Inverse Functions

I. Introductory Concept
Consider your learner reference number (LRN). Do you have a
classmate with the same LRN as yours? What do you think is the reason why
the Philippine government is implementing the national ID system? These are
just some instances where one-to-one correspondence is applicable in your
daily life. In this Learner’s Packet, you are about to learn the different
concepts related to one-to-one functions and inverse functions.

II. Learning Skills from the MELCs

At the end of this Learner’s Packet, you will be able to:

1. represent real-life situations using one-to-one functions (M11GM-Id-1);
2. determine the inverse of a one-to-one function (M11GM-Id-2);
3. represent an inverse function through its: (a) table of values, and (b)
graph (M11GM-Id-3); and
4. find the domain and range of inverse functions. (M11GM-Id-4).

III. Learning Activities

LESSON 1. REPRESENTING REAL-LIFE SITUATIONS USING ONE-TO-ONE

FUNCTIONS

Observe each figure.

X y X y X y
Mrs. Cruz John -2 0
Peter -1 Philippines Manila
Mrs. Lee Oscar 0 1 Japan Tokyo
Ella 1 Indonesia Jakarta
Lyn 2 4 South Korea Seoul
Fig. 1 Mother to her Fig. 2 A number to its Fig. 3 Country to its capital
Children Relation square relation relation

Questions:
1. Which of the figures illustrate a function?

2. What type of function is shown in figure 1? Figure 2?

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DISCUSSION

Based on the illustrations, figures 2 and 3 show functional relationship.

 Figure 1 is not a function since one element in the first column is paired with more
than one element in the 2nd column.

 Figure 2 illustrates many-to-one function since more than one element in the first
column is paired to one element in the 2nd column.

 The third figure shows one-to-one function since each element in the first column is
paired to only one element in the 2nd column.

Definition: The function f is one-to-one if for any x 1 , x 2 in the domain of f, then f(x 1) ≠
f(x2). That is, the same y-value is never paired with two different x-values.

Illustration 1:
In the diagram below, observe that each student is paired to a unique Student ID number.

Name of Student Student’s ID Number

Allan 0315
Ben 0418
Rob 0010
Sam 1035

Since, each 1st element (name of student) is paired to a unique 2nd element (student ID
number). That is, every item from the first set has exactly one partner on the other set. Thus,
this diagram shows one-to-one function.

Illustration 2: Given the set of ordered pairs

A={(1, 2),(2, 3) ,(3 , 4),(4 , 5),(5 , 6)}
Observe that each 1 element x is paired to a unique 2nd element y . Thus, this set
st

represents one-to-one function.

Representing Real Life Situations Using One-to-One Function

In real life, there are many situations that represent one-to-one function. Let’s take a
look at the following examples.

Examples 1-3 are illustrating one-to-one functions and not one-to-one functions.

Example 1. The relation pairing an SSS number to an SSS member.

Solution: Each SSS member is assigned to a unique SSS number. Thus, the relation is a
function. Further, two different members cannot be assigned the same SSS number. Thus,
the function is one-to-one.

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Example 2. The relation pairing a real number to its square.

Solution. Each real number has a unique perfect square. Thus, the relation is a function.
However, two different real numbers such as 2 and –2 may have the same square. Thus, the
function is not one-to-one.

Example 3. The relation pairing an airport to its airport code

Airport codes are three letter codes used to uniquely identify airports around the
world and prominently displayed on checked-in bags to denote the destination of these bags.
Here are some examples of airport codes:
 MNL – Ninoy Aquino International Airport (All terminals)
 CEB – Mactan-Cebu International Airport
 DVO – Francisco Bangoy International (New York City)
 CDG – Charles de Gaulle International Airport (Paris, France)
Airport codes can be looked up at https://www.world-airport-codes.com

Solution. Since each airport has a unique airport code, then the relation is a function. Also,
since no two airports share the same airport code, then the function is one-to-one.

Determining a graph which represents One-to-One and Not One-to-One Function

Given the graph of a function f , a HORIZONTAL LINE TEST can be used to determine if it is
one-to-one or not.

HORIZONTAL LINE TEST

A function is one-to-one if each horizontal line does not intersect the graph at more than
one point.
Example 4. Solution:
If you draw a horizontal line to the
graph, observe that the horizontal line
(solid horizontal line) passes through
two points. Thus, this graph does not
represent one-to-one function.

Example 5.

Solution:
If you draw a horizontal line to
any part of the graph, observe that
the horizontal line (solid horizontal
line) passes through only one point
on the graph. Thus, this graph
represents one-to-one function.

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 ACTIVITY 1: Match One-to-One
______________
Directions: Match one in column A to one in column B to make their
relation as one-to-one function. Write the letter of your answer on your
answer sheet.
Column A Column B
1. Name of Country A. ZIP Code
2. SIM Card B. Motorbike/Vehicle
3. Plate Number C. Official Seal/Logo
4. Name of School D. Mobile Number
5. Name of Agency E. School ID
F. Country Code
ACTIVITY 2: Check One or Cross One?
___________
Directions: Put a check mark if the given relation is one-to-one. Put a cross mark if the
given relation is not a one-to-one function. Write your answer on your answer sheet.

______1. The relation pairing a person to his or her citizenship

______2. The relation pairing a distance d (in kilometers) travelled along a given jeepney
route to the jeepney fare for travelling that distance
______3.

_____ 4. _______ 5.

ACTIVITY 3: _We are the One.

_ ___________
Directions: Write three (3) examples of real-life situations showing one-to-one function.

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LESSON 2. DETERMINING THE INVERSE OF ONE-TO-ONE FUNCTIONS

Let f be a one-to-one function with domain A and range B. Then the inverse of f
denoted f −1is a function with domain B and range A denoted by f −1 ( y )=x if and only if
f (x)= y for any y in B.

Illustration:
Example 1. Let A={(1, 2),(2, 3) ,(3 , 4),(4,5)}. The inverse of A denoted by A−1 can be
obtained by interchanging the values of x and y , that is

A−1={(2,1) ,(3 , 2),(4 , 5),(5,4)}

Example 2. Let B={(−2 ,−4),(−1 ,−2),(0,0),(1 ,2) ,(2 , 4) }, the inverse of B is

B−1 ¿ {(−4 ,−2) ,(−2 ,−1) ,(0,0), (2,1),( 4 , 2)}.

Example 3. Given a table of values, the inverse can be determined by interchanging the
values of x and y .

Let y=x +1 be illustrated in the table

x -2 -1 0 1 2
y -1 0 1 2 3
The inverse of y=x +1 is

x -1 0 1 2 3
y -2 -1 0 1 2

A function has an inverse if and only if it is one-to-one.

Given an equation, the inverse can be determined by following these 3 steps.

1. write the function in the form y=f (x );
2. interchange the x and y variables;
3. solve for y in terms of x .

Example 4. Find the inverse of f (x)=3 x +1

Solution.
a. The equation of the function is y=3 x +1.
b. Interchange the x and y variables: x=3 y +1
c. Solve for y in terms of x :

x−1
Therefore, the inverse of f (x)=3 x +1 is f −1 (x)= .
3
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Example 5. Find the inverse of f (x)= x2 + 4 x – 2.
Solution:
a. the equation of the function is y=x 2 +4 x – 2
b. interchange x and y variables x= y 2 +4 y – 2
c. solve for y in terms of x

The equation y=± √ x+ 6−2 does not represent a function because there are some x -
values that correspond to two different y-values (e.g. if x=3, y can be 1 or – 5.) Therefore,
the function f ( x )=x 2 +4 x – 2 has NO INVERSE FUNCTION.

x−1
Example 6. Find the inverse of f (x)=
x +1
Solution:
x−1
a. the equation of the function is y= .
x +1
y−1
b. interchange x and y variables x=
y +1
c. solve for y in terms of x

y−1
x=
y +1
x ( y +1 ) = y−1(by multiplying( y +1)both sides)
property
xy + x= y−1(by distributive )
addition
xy− y =−x−1(by simplifying like terms )
y ( x−1 )=−x−1(by common factoring)
−x−1
y= (by dividing ( x−1 ) both sides)
x−1

x−1 −1 −x−1
Therefore, the inverse of f (x)= is f ( x )= .
x +1 x−1

Remember the property of an inverse of one-to-one function.

Property of an Inverse of One-to-One Function

Given a one-to-one function f ( x ) and its inverse f −1 ( x ). Then, the following are
true.
(a) The inverse of f −1 ( x ) is f ( x ).
(b) f ( f −1 ( x ) )=x for all x in the domain of f −1.
(c) f −1 ( f ( x ) )=x for all x in the domain of f .

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Verifying these properties is left for you to answer.

Notes:
The function has its inverse:
The f (x)=a xn + b , where a ≠ 0 and n is an odd whole number.
ax +b
The f (x)= , where a ≠ 0, c ≠ 0 and a , b , c∧d are constants.
cx+ d
ACTIVITY 1: Do I have an Inverse?

Directions: Classify the following functions whether they have an inverse or they don’t have.
List the letter of the functions with inverse on the WI column and those without inverse on
the WOI column.
1. E={( 2, 4) ,(3 ,6) ,( 4 , 8) ,(5 ,10) }
2. Q={(0,1) ,(1 , 4) ,(2 ,7) ,(3 , 10)}
3. U ={(−2 ,−8),(0,0) ,(2,8),( 4,64) ,(6,216)}
4. I ={(−2 ,−7),(−1 , 0) ,(0,1),(1,2) ,(−2,9)}
2 1
(
5. P={ −1,−
3)( )
, 0 ,− ,(1 , 0)}
4
WI WOI

ACTIVITY 2: TRUE or FALSE?

Directions: Write the word TRUE if the 2nd function is the inverse of the 1st function; write the
word FALSE if it is not.
X +1
1. y=3 x – 1; y−1=
3
X
2. y=5 x 5 ; y−1=

−1
√
5

5
X +2
3. y=x 2 – 2; y =
2
4. y=x 3 – 1 ; y−1=√3
x+ 1
x +1 −1 3 x +1
5. y= ;y =
2 x−3 2 x−1

ACTIVITY 3: Do I Exist? Find My Inverse!

Directions: Find the inverse of each function, if it exists. If it does not exist, write No Inverse.
1. A={(0 ,−1) ,(1,0) ,(2,3),(3,8) ,(4,15)}
2. B={(−2,5),(0 ,−3),(1 ,−1)}
3. y=7 x 4 – 8
3
4. y= √ 1− x

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 4
5. y=
x−1

LESSON 3: REPRESENTING AN INVERSE FUNCTION THROUGH TABLE OF VALUES

AND GRAPH

In the previous lesson, you were able to find the inverse of a function given an
equation. In this lesson, we will represent the inverse of a function given the table of values
and given a graph.

Representing an Inverse Function through Table of Values

Given a function represented through a table of values, the inverse can be determined
by interchanging the values of x and y .

Example 1. Determine the inverse of the one-to-one function y given the table of values.

x -2 -1 0 1 1.5
y -3 -1 1 3 4
Solution: By interchanging the values of x and y we get y−1

x -3 -1 1 3 4
y−1 -2 -1 0 1 1.5

Example 2. Determine the inverse of the one-to-one function f (x) given the table of values
x 0 1 2 3 4
f (x) 0 1 4 9 16
Solution: By interchanging the values of x and f (x) we get f −1 (x)

x 0 1 4 9 16
−1
f (x) 0 1 2 3 4

Looking at these two examples, given a set of ordered pairs and a table, we can determine
the inverse of one-to-one functions by interchanging the values of x and y .

Representing the Inverse of a Function through a Graph

Given the graph of a one-to-one function, the graph of its inverse can be obtained by
reflecting the graph about the line y=x .
Example 3. Represent the inverse of f (x)=2 x +1whose graph is shown below.

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Solution:
Let us consider the table of values for the graph of f (x)=2 x +1
x -2 -1 0 1 2
f (x) -3 -1 1 3 5
By interchanging the values of x and y in the table above, we get the inverse of
f (x)=2 x +1, whose table is shown below.
x -3 -1 1 3 5
−1
f ( x) -2 -1 0 1 2
Plotting the points on the coordinate plane, we get the graph of the inverse of
f (x)=2 x +1

If we put the graph of f (x)=2 x +1 and its inverse in one coordinate plane, we will
have the figure below. The solid line is the graph of the original function f (x)=2 x +1 while
the dashed line is the graph of its inverse.
f (x)

f −1 ( x)

Observe that if we draw a line y=x (shown by the dotted line), it can be noted that
the graph of the inverse of f (x)=2 x +1 is a reflection of the graph of the original function
through the line y=x .

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 Thus, to represent the graph of the inverse of the given function, we simply draw the
line y=x (shown by the dotted line) and reflect the given graph across the line y=x .

3
Example 4. Represent through graph the inverse of f ( x )= √ x +1 using the given graph.

Solution:
Applying the horizontal line test, we confirm that the function is one-to-one. Reflect
the graph of f (x) across the line y=x to get the plot of the inverse function.
f −1 ( x)

f (x)

ACTIVITY 1: Interchanging My Domain and Range

Directions: Match column A with the corresponding inverse in column B. Write the letter of
your answer on your answer sheet.
COLUMN A COLUMN B
1. A.
X -2 -1 0 1 2 x -5 -2 1 4 7
Y -5 -2 1 4 7 y -2 -1 0 1 2
2. B.
x 1 2 3 4 5 x 0 5 10 15 20
y 1 1/2 1/3 1/4 1/5 y 0 1 2 3 4
3. C.
x 1 1/2 1/3 1/4 1/5
RO_General Mathematics_Grade 11_Q1_LP 4 y 1 2 3 4 5
D. 10
 x 0 1 2 3 4
y 0 5 10 15 20
4.

5.

ACTIVITY 2: Showing My Reflection

Directions: Write the word TRUE if the graph shows inverse functions and FALSE if
otherwise.
1. 2. 3.

4. 5.

ACTIVITY 3: Giving My Inverse

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Directions: Give the inverse of each function.
1. 2. 3.
x -3 -2 -1 0 1
y -24 -5 2 3 4

LESSON 4: FINDING THE DOMAIN AND RANGE OF INVERSE FUNCTIONS

Based on previous lessons, you learned that the domain of a function is the set of all
possible values of x while the range is the set of all possible values of y . Let us recall how
to find the domain and range of a function.

Directions: Find the domain and range of each function.

1. A={(−4 , 4),(−3 ; 2) ,(−2 , 1),( 0 ,−1),( 1,−3),(2,−5)}

Answer: the domain is the list of all 1st elements in the set of ordered pairs while the range is
the list of all 2nd elements in the set of ordered pairs, thus
Domain: {-4, -3, -2, -1, 2}
Range: {4, 2, 1, -1, -3, -5}

2. Table of values
x -2 -1 0 1 2
y 4 1 0 1 4

Answer: the domain is the list of all x elements in the table while the range is
the list of all y elements in the table, thus
Domain: {−2 ,−1 , 0 , 1 ,2 }
Range: {4 , 1 ,0 ,1 , 4 }

3. Graph

Answer: Domain: {x ϵ R∨−2≤ x ≤ 2}

Range: { y ϵ R∨−5 ≤ y ≤ 3}

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 Looking back at the previous lessons, you have learned that the domain of a function
is the set of all possible values of x and the range is the set of all possible of values of y.

The domain of the inverse function is the range of the original function, and
the range of the inverse function is the domain of the original function.

Illustration:
Given f (x)=1 – x where x is restricted to 0 , 1 ,2 , 3.
The table of values is shown below.

x 0 1 2 3
f (x) 1 0 -1 -2

The domain of f (x) is {0, 1,2,3} while the range of f (x) is {1, 0, -1, -2}.

What is the inverse of the function shown in the table above?

 The inverse of the table above is

x 1 0 -1 -2
f (x) 0 1 2 3

What is the domain and range of its inverse?

 The Domain of the inverse of the function above is {1, 0, -1, -2}.
 The Range of the inverse of the function above is {0, 1, 2, 3}.

To sum it up, observe the table below.

f (x)=1 – x where x is 0, 1, 2, 3 Inverse of f (x)

Domain {0, 1, 2, 3} {1, 0, -1, -2}
Range {1, 0, -1, -2} {0, 1, 2, 3}

What do you observe about the domain and range of a function and its inverse?
Let’s try the following examples in determining the domain and range of its inverse.
Example 1. A={(−2 ,−4 ),(−1 ,−2),( 0 ,0) ,(1 , 2) ,(2 , 4) }
Solution: The Domain of A={−2 ,−1 , 0 ,1 , 2} and the Range of
A={−4 ,−2 , 0 , 2, 4 }
Therefore, the Domain of A−1={−4 ,−2 , 0 , 2, 4 } and the Range of A−1={−2 ,−1 , 0 ,1 , 2 }

Example 2. f ( x )=x 2 – 1 restricted to x is -2, 0, 2, 3, 4.

x -2 0 2 3 4
f (x) -9 -1 7 26 63

Solution:The Domain of f (x)={−2 ,0 , 2 , 3 , 4 } and the Range of f (x)={−9 ,−1 , 7 , 26 , 63}

Thus, Domain of f −1 (x)={−9 ,−1 , 7 , 26 , 63} and the Range of f −1 ( x )={−2, 0 , 2 ,3 , 4 }

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Now that you have learned how to determine the domain and range of inverse functions
given a set of ordered pairs and a table, let us now discuss how to determine the domain
and range of inverse functions given the graph of the function.

Given a graph, the domain and range of the inverse of a function can be
determined by inspection of the graph.

Example 3. Find the domain and range of the inverse function of f (x)=2 x +1 restricted in
the domain {x∨– 2 ≤ x ≤1.5 }whose graph is shown below. (the solid line is the graph of f (x)
, the dashed line is the graph of f −1 ( x )∧the dotted line isthe graph of y=x )

Solution:
The domain of f (x) is {x ∈ R l-2 ≤ x ≤ 1.5\}.
The range of f (x) is { y ∈ R l-3 ≤ y ≤ 4\}.

Therefore,
The domain of f −1 ( x) is {x ∈ R l-3 ≤ x ≤ 4\}
The range of f −1 (x) is { y ∈ R l-2 ≤ y ≤ 1.5\}
5 x−1
Example 4. Consider the rational function f ( x )= whose graph is shown below. Find
−x +2
the domain and range of its inverse.

Solution:
From our lessons on rational functions, we get the following results.
Domain of f (x)={x ϵ R∨x ≠ 2 }
Range of f (x)={ y ϵ R∨ y ≠−5 }

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 Therefore,
Domain of f −1 (x)={x ϵ R∨x ≠−5 }
Range of f −1 (x)={ y ϵ R∨ y ≠ 2}

ACTIVITY 1: Pick my inverse’s domain and range

Directions: Find the domain and range of the inverse of each one-to-one function. Pick
the letter of the correct answer from the box below. Write the letter of your answer on your
answer sheet.
1. A={(2, 7) ,(3 , 9),(4 , 11) ,(5 , 13) ,(6 ,15) }
2. B={(−2 ,−6),(−1, 1),(0 ,2) ,(1 ,3),(2 , 10) }
3. f (x)=3 x−1 restricted to x values 0, 1,2,3 and 4
x 0 1 2 3 4
f (x) -1 2 5 8 11
3
4. f (x)= x −2 restricted to x values 0, 1, 2, 3 and 4
x -2 -1 0 1 2
f (x) -10 - 3 -2 -1 6
5. y=4 – 2 x restricted to x = {-1, 0, 1, 2, 3}
x -1 0 1 2 3
y 6 4 2 0 -2

M Domain: { 7, 9, 11, 13, 15} Range: { 2, 3, 4, 5, 6}

A Domain: { 6, 4, 2, 0, -2} Range: { -1, 0, 1, 2, 3}
T Domain: { -10, -3, -2, -1, 6} Range: { -2, -1, 0, 1, 2}
H Domain: { -6, 1,2,3,10} Range: { -2, -1, 0, 1, 2}
S Domain: {-1, 2, 5, 8, 11 } Range: {0, 1, 2, 3, 4 }

ACTIVITY 2: The Domain and Range of the Inverse

of the Graph
Directions: Determine the domain and range of the inverse of each function given the
graph.
1. B={(−2 ,−6),(−1, 1),(0 ,2) ,(1 ,3),(2 , 10) }

Domain of Inverse: __________________

Range of Inverse: ___________________

2. f (x)=3 x−1 restricted to x values 0, 1,2,3 and 4

x 0 1 2 3 4
y -1 2 5 8 11

Domain of Inverse: ______________

Range of Inverse: _________________
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3.

Domain of Inverse: ______________________

Range of Inverse: _______________________

ACTIVITY 3: The Domain and Range of Inverse Function

Directions: Determine the Domain and Range of the Inverse of each function.

1. D={( 0 ,3),(1 ,3 /2) ,(2 , 1),(3 , ¾),( 4 , 3/5)}

2. f (x)=3 – 2 x where x is restricted to {-3, -1, 0, 1, 3}

3.

IV. Rubrics for Scoring

Lesson 1 – Activity 1 – 3 1 point for every correct answer

Lesson 2 – Activity 1 – 2 1 point for every correct answer
Activity 3 2 points for every correct answer with solution
Lesson 3 – Activity 1 – 2 1 point for every correct answer
Activity 3 (table) 1 point for correct values of x and 1 point for correct values of y
Activity 3 (graph) 3 points if the all points are correctly plotted and neatly connected
2 points if all the points are correctly plotted but not neatly connected
1 point if the points are not correctly plotted
Lesson 4 – Activity 1 1 point for every correct answer
Activity 2-3 1 point for the correct domain and 1 point for correct range

V. Reflection

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Directions: Answer the following questions.
1. How did you find the lesson?
____________________________________________________________________
____________________________________________________________________
___________________________________________________________________
2. What are the difficulties you have encountered in understanding the lesson? How did
you solve them? ______________________________________________________
____________________________________________________________________
____________________________________________________________________

VI. Answer Key

LESSON 1
Activity 1 1. F 2. D 3. B 4. E 5. C
Activity 2 1. X 2.  3. X 4.  5. 
Activity 3 Answers may vary.

LESSON 2
Activity 1
WI WOI
E I
Q
U
P
Activity 2 1. TRUE 2. TRUE 3. FALSE 4. TRUE 5. TRUE

Activity 3
1. A−1={ (−1,0 ) , ( 0,1 ) , ( 3,2 ) , ( 8,3 ) , ( 15,4 ) } 2. B−1={(5 ,−2),(−3,0) ,(−1,1)}
−1 4+ x
3. NO INVERSE 4. y−1=1−x 3 5. y =
x

LESSON 3
Activity 1 1.A 2. C 3. B 4. D 5. E
Activity 2 1.TRUE 2. TRUE 3. TRUE 4. TRUE 5. TRUE

Activity 3
1.x -24 -5 2 3 4 2. 3.
y -3 -2 -1 0 1 .

LESSON 4
Activity 1 1. M 2. H 3. S 4. T 5. A
Activity 2
1. Domain: {-6, 1, 2, 3, 10} Range:{-2, -1, 0, 1, 2}
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2. Domain: {-1, 2, 5, 8, 11} Range:{0, 1, 2, 3, 4}
3. Domain: { x ∈ R l – 5 ≤ x ≤ 5} Range:{ y ∈ R l – 2 ≤ y ≤ 3}
Activity 3
1. Domain: . { 3, 3/2, 1, ¾, 3/5} Range:{ 0, 1, 2, 3, 4}
2. Domain: . { 9, 5, 3, 1, -3} Range:{ -3, -1, 0, 1, 3}
3. Domain: . { x ∈ R l x ≠ 1} Range:{ y ∈ R l y ≠ - 2}

VII. References

Castillo, Leticia L. et.al. College Algebra. Mandaluyong City:National Book Store, 2009
General Mathematics Learner’s Material
General Mathematics Teacher’s Guide
Obaña, Generoso G. and Mangaldan, Edna R. Making Connections in Mathematics IV.
Manila: Vicarish Publication and Trading, Inc., 2004
Meriam Webster Dictionary (smartphone app)

DEVELOPMENT TEAM OF THE LEARNING ACTIVITY SHEET

WRITER : MARICRIS B. RICAFRENTE - Calabanga National High School

ILLUSTRATOR : MARICRIS B. RICAFRENTE - Calabanga National High School
REVIEWER : ROGEL JOHN O. NAVAL – Sta. Cruz National High School
EDITOR : SONIA VELITARIO – MORAL - Colacling National High School
LAYOUT ARTIST : JHOMAR B. JARAVATA – Bula National High School
VALIDATORS : MICHELLE B. BALUIS – Juan L. Filipino Memorial High School
: ARTHUR LAWRENCE T. YEE - Nabua National High School
FROILAN R. DOBLON – San Fernando National High School

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