Republic of the Philippines
Department of Education
REGION VIII
DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
FUNCTIONS
Title
I. Introduction
In Mathematics, a function is a relation between a set of inputs and set
permissible outputs. Function has the property that each input is related to exactly
one output.
We can relate functions to a vending Machine. You “input” money and your
“output” is a candy or chips
II. MELC with code
Represents real-life situations using functions, including piece-wise
functions. M11GM-Ia-1
III. Strategy
A. Explore
EXAMPLE: “Multiply by 2” is a very simple function. Answer and complete
the table below
Input
Relationship
Output
0
x2
0
2
x2
4
7
x2
10
x2
12
x2
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50
x2
B. Learn
The previous activity helps us understand/grasp what FUNCTION
means.
Function
A Function is a relation where each element in the domain is related to only
one value in the range by some rule
A Function is a set of ordered pairs (x,y) such that no two ordered pairs
have the same x-value but different y-values. Using function functional notation,
we can write f(x)=y, read as “f of x is equal to y”. In Particular, if (1,2) is an ordered
pair associated with the function f, then we say that f(2)=1
EXAMPLE
1. Which of the following relations are functions?
f = { (1,2), (2,3), (3,5), (4,7)}
g = { (1,3), (1,4), (2,5), (2,6), (3,7)}
h = { (1,3), (2,6), (3,9), (n, 3n),…}
Solution:
The relation f and h are functions because no two ordered pairs have the same x-value but
different y-values. Meanwhile, g is not a function because (1, 3) and (1, 4) are ordered pairs with the
same x-value but different y-values.
Relations and Functions can be represented by mapping diagrams where the elements of the
domain are mapped to the range using arrows. In this case, the relation or function is represented
by the set of all the connections represented by by the arrows.
VERTICAL LINE TEST- A graph represents a function if and only if each vertical line intersects the
graph at most once.
Function
Not Function
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PIECEWISE FUNCTIONS
Some situations can only be described by more than one formula, depending on the value of
the independent variable.
EXAMPLE
1. A user is charged P300.00 monthly for a particular mobile plan, which includes 100 free text
messages. Messages in excess of 100 are charged P1 each. Represent the monthly cost for text
messaging using the function t(m), where m is the number messages sent in a month.
SOLUTION: The cost of text messaging can be expressed by the piecewise
t (m) = {
300
300 + 𝑚
, 𝑖𝑓 0 < 𝑚 ≤ 100
, 𝑚 > 100
2. A jeepney ride costs P8.00 for the first kilometers, and each additional integer kilometer adds
P1.50 to the fare. Use a piecewise function to represent the jeepney’s fare in terms of the distance
(d) in kilometers.
SOLUTION: The input value is distance and the output is the cost of the jeepney fare. If F(d)
represents the fare as a function of the distance, the function can be represented as follows:
F (d) = {
8.00
(8 + 1.5⌊𝑑⌋ )
, 𝑖𝑓 0 < 𝑑 ≤ 4
, 𝑖𝑓 𝑑 > 4
C. Engage
Direction: Apply the Piecewise Function to answer the problem below.
1. A computer shop charges P20.00 per hour (or a fraction of an hour) for the first
two hours and an additional P10.00 per hour for each succeeding hour. Represent
your computer’s rental fee using the function R(t) where t is the number of hours
you spent on the computer
SOLUTION:
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D. Apply
A. Direction: Write F if the relation is a function and write NF if it is not.
1._________
2._________
3._________
4._________
g= {(1,3),(1,4),(2,5),(2,6),(3,7)}
h= {(1,5),(2,4),(1,5),(2,5),(3,7)}
y= {(1,2),(2,3),(3,5),(4,7),(5,8)}
p= {(1,0),(2,4),(5,5),(7,6),(8,7)}
4
10
5._________ p= {(2,3),(2,4),(5,5),( 2 ,6),(8,7)}
6._________ p= {(0,0),(1,0),(0,4),(5,6),(0,7)}
B. Use the piecewise function to answer the problem below
A student from WNHS-SHS charged P200.00 monthly for a mobile internet plan. Included in
his promo is the 50 free text messages to all network. Messages in excess of P50.00 are
charged P1.00 each. Represent the monthly cost for text messaging using the function t(m),
where m is the number of messages sent in a month.
SOLUTION:
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PARANAS, SAMAR
IV. Guide Questions
DIRECTION; Write F if the graph is a function and NF if it is not
7._______
8._______
9. ________
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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10. _________
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WRIGHT NATIONAL HIGH SCHOOL
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PARANAS, SAMAR
LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
EVALUATING FUNCTIONS
Title
I. Introduction
In our introduction to functions lesson, we related functions to a vending
machine. You input money and your output is candy or chips.
We’re going to go back to that visual as we begin evaluating functions. We
are going to “input” a number and our “output” is the answer.
II. MELC with code
Evaluates a function. M11GM-Ia-2
III. Strategy
A. Explore
EXAMPLE: Evaluate the following function functions at x=1.5
1. f (x) = 2x + 1
2. q (x) = 𝑥 2 -2x + 2
3. g (x) = √𝑥 + 1
4. r (x)=
2𝑥+1
𝑥−1
5. F(x) = ⌊𝑥⌋ + 1, where ⌊𝑥⌋ is the greatest integer function.
B. Learn
The previous activity guides and anchored us on how to evaluate a
particular function.
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SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
Evaluating a Function
Evaluating a function means replacing the variable in the function, in this
case x, with a value of from the function’s domain and computing for the result. To
denote that we are evaluating f at a for some a in the domain of f, we write f(a)
EXAMPLE
1.Evaluate the functions q (x) = 𝑥 2 - 2x + 2 at x=3
Solution:
q (x) = 𝑥 2 - 2x + 2
substitute/plug-in the value of x to the equation
q (x) = (3)2 – 2(3) + 2
simply
q (x) = 9 -6 + 2
simplify further (using MDAS)
q (x) = 5
final answer.
2. Evaluate the function t (x) =
2𝑥+1
𝑥−1
at x= (a + 1)
Solution:
t(x)=
2𝑥+1
t(x)=
2(𝑎+1)+1
t(x)=
2𝑎+2+1
t(x)=
2𝑎+3
substitute/plug-in the value of x to the equation.
𝑥−1
simplify 2(a + 1) using distributive property
(𝑎+1)−1
simplify further by combining like terms
𝑎+1−1
Final answer
𝑎
3. Evaluate the function g(x) = 𝑥 2 + 2x +1 at x = b -1
Solution:
g(x) = 𝑥 2 + 2x +1
substitute/plug-in the value of x to the equation.
g(x) = (𝑏 − 1)2 + 2(b – 1) +1
simplify (𝑏 − 1)2 using foil method while 2(b – 1)
using distributive property
g(x) =( 𝑏 2 – 2b + 1)+ 2b – 2 +1
combine like terms
g(x) = 𝑏 2
final answer
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C. Engage
For what values of x can we not evaluate the function f(x) =
SOLUTION:
D. Apply
A. Evaluate the following functions at x = 3
1. f(x) = x - 3
2. g(x) = 𝑥 2 – 3x + 5
3
3. h(x) = √𝑥 2 + 𝑥 3
4. p(x) =
𝑥 2 +1
𝑥−4
5. f(x) = |𝑥 − 5| where |𝑥 − 5| means the absolute value of x – 5.
B. Evaluate the function f(x) = 4𝑥 2 – 3x when f(a + b)
SOLUTION:
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𝑥+3
𝑥 2 −4
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DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
IV. Guide Questions
DIRECTION: Evaluate the following functions at x = -4
1. f(x) = 𝑥 3 – 64
2. g(x) = |𝑥 3 − 3𝑥 2 − 1|
3. r(x) = √5 − 𝑥
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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Wright NHS Senior High Department
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Department of Education
REGION VIII
DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
OPERATIONS ON FUNCTIONS
Title
I. Introduction
Functions with overlapping domains can be added, subtracted, multiplied,
and be divided. If f(x) and g(x) are two functions, then for all x in the domain of both
functions the sum, difference, product and quotient are define as follows
(f + g) (x) = f(x) + g(x)
(f – g) (x) = f(x) – g(x)
(f * g) (x) = f(x) * g(x)
(f / g) (x) = f(x)/g(x)
II. MELC with code
Performs addition, subtraction, multiplication, division, and compositions of
functions.M11GM-Ia-3
III. Strategy
A. Explore
EXAMPLE:
and
2
5
2. Find the difference of
3
6
1.Find the sum of
1
2
and
2
6
3. Find the product of (x-1) and (x + 1)
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B. Learn
Operations on functions ( sum, difference, product, and quotient)
1
𝑥−3
Find the sum of
2
𝑥−5
and
Solution:
1.
1
𝑥−3
+
2
𝑥−5
=
=
=
=
𝑥−5
(𝑥−3)(𝑥−5)
+
2(𝑥−3)
(𝑥−3)(𝑥−5)
𝑥−5+ 2(𝑥−3)
(𝑥−3)(𝑥−5)
𝑥−5+ 2𝑥−6
(𝑥−3)(𝑥−5)
𝑥−11
𝑥 2 −8𝑥+15
2. Find the Product of
Ans.
𝑥 2 −4𝑥−5
𝑥 2 −3𝑥+2
and
𝑥 2 −5𝑥−6
𝑥 2 −3𝑥−10
Solution:
𝑥 2 −4𝑥−5
𝑥 2 −3𝑥+2

=
(𝑥+1)(𝑥−5)
=
(𝑥+1)(𝑥−3)
=
(𝑥−2)(𝑥−1)
𝑥 2 −5𝑥−6
𝑥 2 −3𝑥−10

(𝑥−2)(𝑥−3)
(𝑥−5)(𝑥+2)
(𝑥−1)(𝑥+2)
𝑥 2 −2𝑥−3
𝑥 2 + 𝑥−2
Ans.
2. Find the quotient of
2𝑥 2 +𝑥−6
2𝑥 2 +7𝑥+5
and
𝑥 2 −2𝑥−8
2𝑥 2 −3𝑥−20
Solution:
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=
=
=
=
2𝑥 2 +𝑥−6
÷
2𝑥 2 +7𝑥+5
2𝑥 2 +𝑥−6

2𝑥 2 +7𝑥+5
(2𝑥−3)(𝑥+2)
(2𝑥+5)(𝑥+1)
2𝑥−3
𝑥+1
𝑥 2 −2𝑥−8
Get the reciprocal of
2𝑥 2 −3𝑥−20
𝑥 2 −2𝑥−8
𝑥 2 −3𝑥−20
2𝑥2 −3𝑥−20
𝑥2 −2𝑥−8

(𝑥−4)(2𝑥+5)
(𝑥+2)(𝑥−4)
Ans.
C. Engage
Find the Product
𝑥 2 +4𝑥−21
𝑥 2 −16
( 𝑥 2 −6𝑥+9 ) (𝑥 2 +3𝑥−28 )
SOLUTION:
D. Apply
Consider the following functions below:
g(x)=2x-4
f(x)=x+6
𝑥+5
h(x)= 𝑥+7
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𝑥−3
t(x)= 𝑥+4
Republic of the Philippines
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Determine the following function.
1. (g + f)(x)
2. (h - f)(x)
3. (t  h)(x)
4. (g  f)(x)
5. (f + t)(x)
IV. Guide Questions
𝑥 2 −1
Find the quotient of (
𝑥 2 +4𝑥+4
) and (
𝑥 2 +8𝑥+7
2𝑥+4
)
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
Call or text.
For updates. Visit.
0917-304-5347 / 0949-994-2133
Wright NHS Senior High Department
Republic of the Philippines
Department of Education
REGION VIII
DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
COMPOSITION ON FUNCTIONS
Title
I. Introduction
In our introduction to functions lesson, we related functions to a vending
machine. You input money and your output is candy or chips.
We’re going to go back to that visual as we begin evaluating functions. We
are going to “input” a number and our “output” is the answer.
II. MELC with code
Evaluates a function. M11GM-Ia-2
III. Strategy
A. Explore
EXAMPLE: Evaluate the following function functions at x=1.5
1. f (x) = 2x + 1
2. q (x) = 𝑥 2 -2x + 2
3. g (x) = √𝑥 + 1
4. r (x)=
2𝑥+1
𝑥−1
5. F(x) = ⌊𝑥⌋ + 1, where ⌊𝑥⌋ is the greatest integer function.
B. Learn
The previous activity guides and anchored us on how to evaluate a
particular function.
Call or text.
For updates. Visit.
0917-304-5347 / 0949-994-2133
Wright NHS Senior High Department
Republic of the Philippines
Department of Education
REGION VIII
DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
Evaluating a Function
Evaluating a function means replacing the variable in the function, in this
case x, with a value of from the function’s domain and computing for the result. To
denote that we are evaluating f at a for some a in the domain of f, we write f(a)
EXAMPLE
1.Evaluate the functions q (x) = 𝑥 2 - 2x + 2 at x=3
Solution:
q (x) = 𝑥 2 - 2x + 2
substitute/plug-in the value of x to the equation
q (x) = (3)2 – 2(3) + 2
simply
q (x) = 9 -6 + 2
simplify further (using MDAS)
q (x) = 5
final answer.
2. Evaluate the function t (x)=
2𝑥+1
𝑥−1
at x= (a + 1)
Solution:
t(x)=
2𝑥+1
t(x)=
2(𝑎+1)+1
t(x)=
2𝑎+2+1
t(x)=
2𝑎+3
substitute/plug-in the value of x to the equation.
𝑥−1
simplify 2(a + 1) using distributive property
(𝑎+1)−1
simplify further
𝑎+1−1
Final answer
𝑎
3. Evaluate the function g(x) = 𝑥 2 + 2x +1 at x = b -1
Solution:
g(x) = 𝑥 2 + 2x +1
substitute/plug-in the value of x to the equation.
g(x) = (𝑏 − 1)2 + 2(b – 1) +1
simplify (𝑏 − 1)2 using foil method while 2(b – 1)
using distributive property
g(x) =( 𝑏 2 – 2b + 1)+ 2b – 2 +1
combine like terms
g(x) = 𝑏 2
final answer
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PARANAS, SAMAR
C. Engage
For what values of x can we not evaluate the function f(x) =
SOLUTION:
D. Apply
A. Evaluate the following functions at x = 3
1. f(x) = x - 3
2. g(x) = 𝑥 2 – 3x + 5
3
3. h(x) = √𝑥 2 + 𝑥 3
4. p(x) =
𝑥 2 +1
𝑥−4
5. f(x) = |𝑥 − 5| where |𝑥 − 5| means the absolute value of x – 5.
B. Evaluate the function f(x) = 4𝑥 2 – 3x when f(a + b)
SOLUTION:
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𝑥+3
𝑥 2 −4
Republic of the Philippines
Department of Education
REGION VIII
DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
IV. Guide Questions
DIRECTION: Evaluate the following functions at x = -4
1. f(x) = 𝑥 3 – 64
2. g(x) = |𝑥 3 − 3𝑥 2 − 1|
3. r(x) = √5 − 𝑥
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
Call or text.
For updates. Visit.
0917-304-5347 / 0949-994-2133
Wright NHS Senior High Department
Republic of the Philippines
Department of Education
REGION VIII
DIVISION OF SAMAR
DISTRICT OF WRIGHT I
WRIGHT NATIONAL HIGH SCHOOL
SENIOR HIGH SCHOOL DEPARTMENT
PARANAS, SAMAR
LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
OPERATIONS ON FUNCTIONS
Title
I. Introduction
Let f and g be functions. The Composite Function denoted by (f ◦ g) is
defined by (f ◦ g)(x) = f(g(x)). The process of obtaining a composite function is called
function Composition.
II. MELC with code
Solve problems involving functions. M11GM-Ia-4
III. Strategy
A. Explore
Use the following functions below:
f (x) =2x + 1
g (x) = 𝑥 2 – 1
Find and simplify (g◦ f)(x)
Solution:
(g◦ f)(x) = g(f(x)) = 𝑥 2 -1
= (𝑓(𝑥))2 – 1
= ( 2𝑥 + 1)2 – 1
= 4𝑥 2 + 4x + 1 – 1
= 4𝑥 2 + 4x
(g◦ f)(x) = g(f(x)) = 4x (x + 1) Ans.
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B. Learn
Composition of Functions
Use the following functions below
f (x) = 2x + 1
g(x) = √𝑥 + 1
q (x) = 𝑥 2 – 2x + 2
p(x) =
F(x) = ⌊𝑥⌋ + 1
1. Find and simplify (g◦ f)(x)
Solution:
(g◦ f)(x) = g(f(x)) = √𝑥 + 1
= √𝑓(𝑥) + 1
= √(2𝑥 + 1) + 1
(g◦ f)(x) = g(f(x)) = √2𝑥 + 2 Ans.
2. Find and simplify (q◦ f)(x)
Solution:
(q◦ f)(x) = q(f(x)) = 𝑥 2 – 2x + 2
= (𝑓(𝑥))2 – 2(f(x)) + 2
= (2𝑥 + 1)2 – 2(2x + 1) + 2
= 4𝑥 2 + 4x + 1 - 4x -2 + 2
(q◦ f)(x) = q(f(x)) = 4𝑥 2 + 1 Ans.
3. Find and simplify (f◦ p)(x)
Solution:
(f◦ p)(x) = f(p(x)) = 2x + 1
= 2p(x) + 1
= 2[
= [
=
(f◦ p)(x) = f(p(x)) =
2𝑥+1
𝑥−1
]+1
4𝑥+2
𝑥−1
]+1
(4𝑥+2)+(𝑥−1)
𝑥−1
5𝑥+1
𝑥−1
Ans.
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2𝑥+1
𝑥−1
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C. Engage
Let f (x) = 3𝑥 2 – 2x – 1
Solve and simplify
and h (x) = 𝑥 2 – 1.
(f◦ h)(x)
SOLUTION:
D. Apply
Let t (x) =
𝑥+1
and
𝑥−1
k (x) = 𝑥 2 – 1.
Solve and simplify
(k◦ t)(x)
Solution:
IV. Guide Questions
Let f(x) = 2x – 15
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Determine f(f(f(f(15))))
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
REPRESENTING REAL-LIFE SITUATIONS USING
RATIONAL FUNCTIONS
Title
I. Introduction
A polynomial function p of degree n is a function that can be written in
the form p (x) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−2 𝑥 𝑛−2 + …+ 𝑎1 𝑥 𝑛 + 𝑎0 where 𝑎0 , 𝑎1 ,…, 𝑎𝑛 ϵ IR,
𝑎𝑛 ≠ 0, and n is a positive integer. Each addend of the sum is a term of the
polynomial function. The constant 𝑎0 , 𝑎1 , 𝑎2 ,…𝑎𝑛 are the coefficients. The leading
coefficient is 𝑎𝑛 . The leading term is 𝑎𝑛 𝑥 𝑛 , and the constant term is 𝑎0 .
II. MELC with code
Represents real-life situations using rational functions. M11GM-Ib-1
III. Strategy
A. Explore
An object is to travel a distance of 10 meters. Express velocity v as a function of
travel time t, in seconds
The following table of values show v for various of t.
t (seconds)
v (meter per second
1
2
3
4
5
10
5
2.5
2
1
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The function v (t) =
10
𝑡
can represent v as a function of t.
B. Learn
A Rational Function is a function of the form f(x) =
𝑝(𝑥)
𝑞(𝑥)
where p(x) and q(x) are
polynomial functions and q(x) is not zero function (i.e., q(x) ≡ 0). The domain of f(x)
is the set of all values of x where q(x) ≠ 0.
EXAMPLE
5𝑡
(in mg/mL) represents the concentration of a drug in a
patient’s bloodstream t hours after the drug was administered. Construct a
table of values for c(t) for t = 1, 2, 5, 10. Round off answers to three decimal
places. Use the to sketch a graph and interpret the results.
Suppose that c(t) =
𝑡 2 +1
Solution:
a. c(0) =
5𝑡
𝑡 2 +1
b. c(1) =
c. c(2) =
d. c(5) =
=
5(1)
(1)2 +1
5(2)
(2)2 +1
5(5)
(5)2 +1
e. c(10) =
5(0)
(0)2 +1
= 2.5
=2
= 0.962
5(10)
(10)2 +1
t
C(t)
=0
= 0.495
0
0
1
2.5
2
2
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5
0.962
10
0.495
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C. Engage
In an organ pipe, the frequency f of vibration of air is inversely proportional to the
length L of the pipe. Suppose that the frequency of vibration in a 10-foot pipe is 54
vibrations per second. Express f as a function of L.
Solution:
Since f is inversely proportional to L, then f =
proportionality.
If L= 10 then f=54. Thus, 54 =
as a function of L.
𝑘
,
10
𝑘
𝐿
, where k is the constant of
k= 540. Thus, the function f(L)=
D. Apply
Given the polynomial function p(x) = 12 + 4x - 3𝑥 2 - 𝑥 3 .
1. The degree of the polynomial
Ans.
2. The leading coefficient
Ans.
3. The constant term
Ans.
4. The number of zeroes
Ans.
IV. Guide Questions
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540
𝐿
represents f
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V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
RATIONAL FUNCTIONS, EQUATIONS, AND INEQUALITIES
Title
I. Introduction
A Rational Expression is an expression that can be written as a ratio of two
polynomials.
Some examples of rational expressions are
𝑥
2
,
𝑥 2 +2𝑥+3
5
, 𝑥−3
𝑥+1
II. MELC with code
Distinguishes rational function, rational equation, and rational inequality. M11GMIb-2
III. Strategy
A. Explore
 Every polynomial function is a rational function ex. q(x) = 1
 A function that cannot be written in the form polynomial, ex f(x)=sin(x)
 A constant function such as f(x)= 𝜋
B. Learn
A Rational Function is a function of the form f(x) =
𝑝(𝑥)
𝑞(𝑥)
where p(x) and q(x)
are polynomial functions and q(x) is not zero function (i.e., q(x) ≡ 0). The domain of
f(x) is the set of all values of x where q(x) ≠ 0.
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EXAMPLE
The definitions of rational equations, inequalities, and functions are shown below
Rational equation
Rational Inequality
Rational Function
Definition
An equation
An inequality
A function of the
𝑝(𝑥)
involving rational
involving rational
form f(x) = 𝑞(𝑥)
expression
expressions.
where p(x) and q(x)
are polynomial
functions and q(x)
is not zero function
(i.e., q(x) ≡ 0).
2
3
1
5
2
𝑥 2 +2𝑥+3
Example
=5
<
f (x) =
x 2𝑥
x−4
𝑥
𝑥+1
C. Engage
Determine whether the given is a rational function, rational equation, a rational
inequality or none of these. Explain your answer.
𝑥+ 3
𝑥−5
>√4𝑥 + 7
Ans.___________________________________________
D. Apply
Determine whether the given is a rational function, rational equation, a rational
inequality or none of these.
1. 8 =
𝑥+2
𝑥−1
2. x >√𝑥 + 2
3. 2x ≥
7
𝑥+4
___________________
___________________
____________________
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4. f (x) =
5.
𝑥
2
=
𝑥+ 3
𝑥2 −5
– 6 ____________________
4
𝑥+9𝑥3
____________________
IV. Guide Questions
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
SOLVING RATIONAL EQUATIONS AND INEQUALITIES
Title
I. Introduction
A rational equation or inequality can be solved for all x values that satisfy
the equation or inequality. This lesson will help students recall and use their
knowledge on finding the LCD, simplifying complex rational expressions and
equations a inequality symbols.
II. MELC with code
Solves rational equations and inequalities. M11GM-Ib-3
III. Strategy
A. Explore
Recall your knowledge on Least Common Denominator, Factorization, simplest
form and answer the problem below. You may solve the unknown variable.
1.
2
5
2.
3
10
+
3
15
=
3.
𝑥 2 −1
𝑥+1
=5
+ x -2 = 2x -3
B. Learn
To solve a rational equation
1. Eliminate denominators by multiplying each term of the equation by the least
common denominator.
2. Note that eliminating denominators may introduce extraneous solutions. Check
the solutions of the transformed equations with the original equation.
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EXAMPLE 1
Solve for x :
2
𝑥
-
3
2x
=
1
5
Solution: The LCD of all the denominators is 10x. Multiply both sides of the
equation by 10x and solve the resolving equation.
2
3
1
10x (𝑥) – 10x(2𝑥) = 10x (5)
20 – 15 = 2x
5 = 2x
5
2
= x or
x=
5
2
Ans.
EXAMPLE 2
Solve for x :
𝑥
𝑥+2
-
1
𝑥−2
=
8
𝑥 2 −4
Solution: Factor each denominator in the rational expression.
𝑥
𝑥+2
-
1
𝑥−2
=
8
(𝑥+2)(𝑥−2)
Multiply the LCD both sides of the equation to remove the denominators.
𝑥
1
8
(x + 2)(x – 2) (𝑥+2) (𝑥−2) = (x + 2)(x – 2)((𝑥+2)(𝑥−2))
x(x – 2) – (x + 2) = 8
𝑥 2 – 3x – 10 = 0
Upon reaching this this step, we can use strategies for solving polynomial
equations.
𝑥 2 – 3x – 10 = 0
(x + 2)(x – 5) = 0
x+2=0
x–5=0
x = -2
x=5
since x = -2 makes the original equation undefined, x = 5 is the only answer.
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To solve rational inequalities
a. rewrite the inequality as a single rational expression on one side of
inequality symbol and 0 on the other side.
b. determine over what intervals the rational expression takes on positive
and negative values.
i. locate the x values for which the rational expression is zero or
undefined(factoring the numerator and denominator is a useful strategy).
ii. Mark the numbers found in (i) on a number line. Use a shaded circle to
indicate that the value is included in the solution set, and a hollow circle to
indicate that the value is excluded. These numbers partition the number
line into intervals.
iii. Select a test point within the interior of each interval in (ii). The sign of
the rational expression at this test point is also the sign of the rational
expression at each interior point in the aforementioned interval.
vi. Summarize the intervals containing the solutions.
EXAMPLE
Solve the inequality
2𝑥
𝑥+1
≥0
Solution:
a. Rewrite the inequality as a single rational expression.
2𝑥
𝑥+1
–1≥0
2𝑥−(𝑥+1)
𝑥+1
𝑥−1
𝑥+1
≥0
≥0
b. The rational expression will be zero at x = 1 and undefined for x = -1. The
value of x = 1 is included while x = -1 is not. Mark these on the number line.
Use a shaded circle for x = 1 ( a solution) and an unshaded circle for x = -1
(not included).
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c. Choose convenient test points in the intervals determined by -1 and 1 to
𝑥−1
determine the sign of 𝑥+2 in these intervals. Construct a table of signs as
shown below.
Interval
x<1
-1<x<1
x>1
Test Point
x = -2
x=0
x=2
x–1
–
–
+
x+1
–
–
+
𝑥−1
𝑥+1
+
–
+
d. Since we are looking for the intervals where the rational expression is
positive or zero, we determine the solution to be the set
{𝑥 𝜖𝑅|𝑥 < −1 𝑜𝑟 𝑥 ≥ 1}. Plot this on the number line.
C. Engage
Solve x :
𝑥2
𝑥−3
=
𝑥+2
2𝑥−5
Solution:
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D. Apply
1. Solve the inequality
2. Solve the inequality
3
𝑥−2
5𝑥
𝑥−1
≥
1
𝑥
<4
IV. Guide Questions
Solve for x :
𝑥+6
𝑥−4
≥
1
𝑥+1
V. Rubrics
VI. References:
GENERAL MATHEMATICS learner’s Material(pg.
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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LEARNER ACTIVITY SHEET/WORKSHEETS in GENERAL
MATHEMATICS
Name of Learner: __________________________
Date: _____________
Grade Level: _________
Teacher_________________________
Section:________
REPRESENTATION OF RATIONAL FUNCTIONS
Title
I. Introduction
A Rational Function is a function of the form f(x) =
𝑝(𝑥)
𝑞(𝑥)
where p(x) and q(x)
are polynomial functions and q(x) is not zero function (i.e., q(x) ≡ 0). The domain of
f(x) is the set of all values of x where q(x) ≠ 0.
Average speed (or velocity) can be computed by the formula s = d/t. consider
a 100-meter track used for foot races. The speed of a runner can be computed by
100
taking time for him to run track and applying it to the formula s =
, since the
𝑡
distance is fixed at 100 meters.
II. MELC with code
Represents a rational function through its: (a) table of values, (b) graph, and (c)
equation. M11GM-Ib-4
III. Strategy
A. Explore
Consider the example below:
Represent the speed of a runner as a function of the time it takes to run 100
meters
Solution:
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Since the speed of a runner depends on the time it takes to run 100 meters, we can
represent speed as a function of time.
Let x represent the time it takes to run 100 meters. Then the speed can be
represented as a function s(x) as follows:
s (x) =
100
𝑥
Observe that it is similar to the structure to the formula s =
distance, and time.
𝑑
𝑡
relating speed,
B. Learn
To further explain the lesson, do consider the second example below
Continuing the scenario above, construct a table of values for speed of runner
against different run times.
Solution:
A table of values can help us determine the behaviour of a function as the variable
x changes.
The current word record ( as of October 2015) for the 100-meter dash run is 9.58
seconds set by the Jamaican Usain Bolt in 2009. We start our table of values at 10
seconds.
Let x be the runtime and s(x) be the speed of the runner in meters per second,
100
where s(x) = 𝑥 . The table of values for run times from 10 seconds to 20 seconds is
as follows:
x
10
12
14
16
18
20
s (x)
10
8.33
7.14
6.25
5.56
5
From the table above we can observe that the speed decreases with time. We can
use a graph to determine if the points on the function follow a smooth curve or a
straight line.
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C. Engage
Plot the points on the table of values on a Cartesian plane. Determine if the points
100
on the function s(x) =
follow a smooth curve or a straight line.
𝑥
Solution:
D. Apply
𝑥−1
1. Represent the rational function given by f(x) = 𝑥+1 using a table of values
and plot a graph of the function by connecting points.
Solution:
IV. Guide Questions
𝑥 2 −3𝑥−10
Represent the rational function given by f(x) =
using a table of values
𝑥
and plot a graph of the function by connecting points.
V. Rubrics
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VI. References:
GENERAL MATHEMATICS learner’s Material(pg.
VII. Key to Correct Answers
Prepared by
TEODORO G. ABAN
Subject Teacher
Noted:
MIGUEL P. DABUET
Secondary School Principal IV
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