Grades 1 to 12 School SAN LUIS NATIONAL HIGH SCHOOL Grade Level 11 – ABM /11 – HUMSS

Daily Lesson Log Teacher NI-KHE G. MARQUEZ Learning Area GENERAL MATHEMATICS
Teaching Dates and Time AUGUST 26 – 29, 2024 Quarter 1st Quarter (WEEK 5)
11 – ABM (7:30 – 8:30 AM)
11 – HUMSS (10:50 – 11:50 AM)

Session 1 Session 2 Session 3 Session 4

I. OBJECTIVES
A. Content Standard The learner demonstrates The learner demonstrates understanding of The learner demonstrates
The learner demonstrates
understanding of key concepts of key concepts of inverse functions, understanding of key concepts of
understanding of key concepts of
inverse functions, exponential exponential functions, and logarithmic inverse functions, exponential functions,
rational functions.
functions, and logarithmic functions. functions. and logarithmic functions.
B. Performance Standard The learner is able to apply the The learner is able to apply the
The learner is able to apply the concepts of
The learner is able to accurately concepts of inverse functions, and concepts of inverse functions, and
inverse functions, and logarithmic functions
formulate and solve real-life problems logarithmic functions to formulate and logarithmic functions to formulate and
to formulate and solve real-life problems
involving rational functions. solve real-life problems with precision solve real-life problems with precision
with precision and accuracy.
and accuracy. and accuracy.
C. MELC and MELC No. M11GM-Ic-3 M11GM-Id-1 M11GM-Id-2 M11GM-Id-3
The learner represents real-life The learner determines the inverse of a The learner represents an inverse
The learner solves problems involving situations using one-to-one functions. one-to-one function. function through its: a) table of values,
rational functions, equations and and b)graph.
inequalities. M11GM-Id-4
The learner finds the domain and range
of an inverse function.
II. CONTENT
III. LEARNING RESOURCES
A. References
1. Teacher’s Guide pages TG for SHS General Mathematics, pp. TG for SHS General Mathematics, pp. TG for SHS General Mathematics, pp. TG for SHS General Mathematics, pp.
43-49 33-42 69-75 69-75
2. Learner’s Material pages LM in General Mathematics, pp. 35-43 LM in General Mathematics, pp. 27-33 LM in General Mathematics, pp. 62-66 LM in General Mathematics, pp. 67-75
3. Textbook pages General Mathematics by Orlando General Mathematics by Orlando Oronce General Mathematics by Orlando
Oronce Series 2016 Series 2016 Oronce Series 2016
4. Additional Material from Teacher’s Guide and Learner’s Material Teacher’s Guide and Learner’s Material Teacher’s Guide and Learner’s Material Teacher’s Guide and Learner’s Material
Learning Resource (LR) Portal
B. Other Learning Resources General Mathematics, Diwa General Mathematics, Diwa Publishing, General Mathematics, Diwa Publishing,
Publishing, Senior High School Series. Senior High School Series. 2016 Senior High School Series. 2016
2016
IV. PROCEDURES
A. Reviewing previous lesson or Recall the steps in solving word Recall rational expressions and rational Recalling the table of values for the Recall the properties of an Inverse
presenting the new lesson. problems in mathematics. equations. These can be useful tools function given by the equation by the Function:
representing real-life situations and for equation y = 2x -1 given below:
1.Read the problem carefully. finding answers to real problems. In Properties of an inverse of a one-to-
2.Identify and list the facts. particular, they are quite good for one function
3.Figure out exactly what the problem describing distance-speed-time
 Given a one-to-one function
f(x0 and its inverse f −1 ( x ) .
Then the following are true:
−1
is asking for. Verify that it is a one-to-one function by a) The inverse of f ( x ) is
4. Find or develop a formula. questions, and modeling multi-person showing that no two y-values share the f(x).
5. Solve the problem. work problems. same x-value. Let us invert the values for x
and y: b) f( f −1 ( x ) ) = x for all x in the
6.Verify the answer.
−1
domain of f .

c) f −1 ¿ = x for all x in the

domain of f.
B. Establishing a purpose for the new
lesson
Complete the drawings by sketching
the reflection of the figure across the
line y = x. For an accurate figure, fold
the paper across the line y = x and trace
Ask: Do you have any idea what is a Ask: Does the table still represent a the figure underneath.
one-to-one function? function?
Definition of one-to-one function is We should see that it can still represent a See attachment (Activity Sheet)
presented. function because each x value is
Connecting the real world scenario and associated with only one y value.
Definition: The function f is one-to-one Elicit from the students the following
the solving of rational Then consider the next table of values.
equations/inequalities. if any x 1 , x 2 ∈the domain of f ,then f ideas:
(x 1)≠ f ¿). That is, the same y –value  The coordinates of the points
is never paired with two different x- in the reflected image are
values. reversed in relation to those in
the original image.
 If a function is not one-to-one,
the reflection of its graph about
the line y = x does not produce
a function.
,
C. Presenting Examples/ instances of Rational equation is useful in solving Ask the class to determine whether the Consider another table of values for
the new lesson given relation is a function. If it is a another function.
problems involving work, motion, and Say: The coordinates of points in a
function, determine whether it is one-to-
others. Let us try the following one. Discuss each. graph of the inverse of a one-to-one
Examples: function are the coordinates of points in
examples.
1.The relation pairing an SSS member the original function with the x and y-
to his or her SSS number. Show the that the table does not represent
coordinates .
(Function/one-to-one) a function because there are some y-
2. The relation pairing a real number to values because here are some y-values Give the Relationship between the
its square. (Function /NOT one-to-one) that are paired with more the one x-value. graphs of a function and its Inverse
3. The relation pairing distance d (in For example, y =1 is paired with x =
 1,2,3,4. Invert the values for x and y. Will
the resulting table still represent a
function?

Graphing Inverse Functions

kilometers) traveled along a given The resulting table does not represent a Given the graph of a one-to-one
jeepney route to the jeepney fare for function since x = 1 is paired with more function, the graph of its inverse can be
traveling that distance. (Function/not than one y-value namely, 1,2,3 and 4. obtained by reflecting the gaph about
one-to-one) the line y = x.
Let students give their own examples of ‘Inverting’ Functions
real-life situations involving functions. The previous discussion shows that
 if the x- and y-values of a one-to-
one function are interchanged,
the result is a function, but
 if the x- and y-values of a function
that is not one-to-one are
inverted, the result is no longer a
function.

D. Discussing new concepts and Graphing one-to-one functions. Define the inverse of a one-to-one
practicing new skills #1 Show the following graphs to the class. function. Given y = f −1 (x) if the graph of y =
(Applications involving Work) Ask them if the given function is one-to-
one or not by having them check if Definition. f(x) = 2x + 1 restricted in the domain
1. Steve and Janet are going to paint
the fence that surrounds their
there is a y-value with more than one Let f be a one-to-one function with { x|−2 ≤ x ≤1.5 } is given below. What
corresponding x-value. domain A and range B. Then the inverse
house today. Steve can paint the is the range of f(x)? What is the domain
Present the graphs. of f, denoted f −1, is a function with domain
fence in 12 hours. Janet can paint and range of its inverse?
the fence alone in 9 hours. How B and range A defined by f −1 ( y )=x if
a) y = x 2−4
long will it take them to paint the and only f(x) = y for any y in B.
fence together. b. y = 2x - 1
Answer:
1 1 t t
t+ t = 1 or + =1
12 9 12 9
Let’s solve the equation for t:
t t
+ =1
12 9
t t
36( + ¿ = 1(36)
12 9
3t +4t = 36
 7t = 36
T = 36/7 or 5.15

Processing of the answer.

To find an equation that represents the

given scenario. Let’s determine how
much of the fence Steve and Janet
paint respectively.

Steve takes 12 hours to paint the fence,

1
each hour he paints .
12 2 x +3
c. y =
x
Since it takes them t hours to paint the 3
fence together. Steve paints 1/12t of d. y=x Solution: Take the reflection of the
the fence. Since it takes Janet 9 hours restricted graph of y = 2x + 1 across the
to paint the fence, each hour she paints line y = x.
1/9 of the fence. Since it takes them t
hours to paint the fence together, Janet
paints 1/9t of the fence.

Since the 1/12 t of the fence that Steve

paints and the 1/9t of the fence that
Janet paints together constitute one
complete fence, we obtain the equation

The range of the original function can be

determined by inspection of the graph.
The range is { y ∈ R|−3 ≤ y ≤ 4 }.
Verify using techniques that the inverse
−1 x−1
function is given by f ( x )= .
2
The students with the teacher’s Continue discussing:
guidance should identify that the The domain and range of the inverse
second, third and fourth graphs function can be determined by
represent one-to-one functions. The inspection of the graph:
first graph is a quadratic function. It is
not one-to-one because the value 0 is
Domain of f
−1
( x ) = [ −3 , 4 ] Range of
paired with two x-values, namely 2 and
 f
−1
( x ) = [ −2 ,1.5 ]In summary,

-2.
A simple way to determine if a given
graph is that of a one-to-one function is
by using the horizontal line test.
Observe that the domain of the inverse
is the range of the original function, and
that the range of the inverse is the
domain of the original function.
E. Discussing new concepts and EXAMPLE 2: SOLVING AN APPLIED Presents the horizontal line test. To determine the inverse of a function from Ask: Is it true for all one-to-one
practicing new skills #2 PROBLEM INVOLVING A RATIONAL A function is one-to-one if each its equation. functions and their inverse?
FUNCTION horizontal line does not intersect the The inverse of the function can be
graph at more than one point. interpreted as the same function but in the Do example 2.
A large mixing tank currently contains opposite direction, that is, it is a function
100 gallons of water into which 5 from the y-value back to its corresponding Example 2.
pounds of sugar have been mixed. A x-value.
tap will open pouring 10 gallons per To find the inverse of a one-to-one 1
Find and graph the inverse of f(x) = ,
minute of water into the tank at the function. x
same time sugar is poured into the tank a) Write the function in the form
whose graph is shown below.
at a rate of 1 pound per minute. Find y=f(x);
the concentration (pounds per gallon) of b) Interchange the x and y variables;
sugar in the tank after 12 minutes. Is c) Solve for y in terms of x.
that a greater concentration than at the
beginning? This is because we are interchanging
the input and output values of a
function.
Giving example.
1. Find the inverse of f(x) = 3x +1.
Solution. The equation of the function
is y = 3x +1. Interchange the x and y
variables: x =3y + 1.
Solve for y in terms of x:
Solution:
x = 3y + 1
x – 1 = 3y
x−1 x−1
=y⟹ y= Solution.
3 3
Therefore, f (x) = 3x +1 is f −1(x) = Applying the horizontal line test, we
x−1 verify that the function is one-to-one.
.
3
Ask the following questions to the class:
 1
Since the graph of f(x) = is
x
symmetric with respect to the line y = x
(indicated by a dashed line), its
reflection across the line y = x is
itself. Therefore the inverse of f(x) is
itself or f −1 ( x )=f ( x ) .

a) What is the inverse of the

inverse?
b) What is f( f −1 ( x) ¿ ? How about
−1
f (f(x))?
Have the class do these on the
example above. The discuss the
following properties that the class
should have observed from the
example above.

Verify that f
−1
( x )=f ( 1x ) using the

techniques in the previous discussion.

(Subtask 2 the following day)

F. Developing mastery (Leads to Discussion. Ask the students to answer activity 3 on
Formative Assessment 3) The first graph showing the plot y = x 2 page 29. (Letter b)
– 4 fails the horizontal line test because
some lines intersect the graph at more See attachment.
than one point. The remaining three
graphs pass the horizontal line test,
because all horizontal lines intersects
the graph at most once.
G. Finding practical applications of
Let the students cite their own example
Cite a real-life situation showing an inverse Give a function f that represents the
concepts and skills in daily living of one-to-one function. price of the apple if a P25.00 price
where rational functions, equations and
inequalities are involved. reduction applies.
See attachment.
H. Making generalizations and Let the learner apply the concept in Instill in the minds of the learners that When to use functions in solving real life
abstractions about the lesson rational equalities and inequalities to real-life situations can all be problems?
 Ask: Enumerate the properties of finding
the inverse of one-to-one function.
represented by functions; specifically
solve word problems. rational functions, rational equations or How to find the inverse of a one-to-
rational inequalities. one function?

I. Evaluating learning A. Enumerate the properties of Short quiz

one-to-one functions.
B. Determine whether the given
Solve the problem. situation is a one-to-one
function or not.
In an inter-barangay basketball
league, the team from Barangay 1. The relation pairing an airport to its
Nabago has won 12 out of 25 airport code. (Function/ one-to-one)
games, a winning percentage of See attachment.
2. The relation pairing a person to his or
48%. How many games should her citizenship. (Not a function)
they win in a row to improve their 3. Books to authors (Function/one-to-
win percentage to 60%? one)
4. SIM cards to cell phones (Not one-to-
one)
5. True or false questions to answers
(Not one-to-one)
J. Additional activities for application Ask students to give one example of a
or remediation situation showing one-to-one function
and one example showing not one-to-
one function. (answers may vary)
V. REMARKS
VI. REFLECTION
A. No. of learners who earned 80% on A. ____ No. of learners who earned A. ____ No. of learners who earned A. ____ No. of learners who earned 80% in A. ____ No. of learners who earned
the formative assessment 80% in the evaluation 80% in the evaluation the evaluation 80% in the evaluation
B. No. of learners who scored below B. ____ No. of learners who require B. ____ No. of learners who require B. ____ No. of learners who require B. ____ No. of learners who require
80% (needs remediation) additional activities for remediation additional activities for remediation additional activities for remediation additional activities for remediation
C. No. of learners who have caught up C. Did the remedial lessons work? C. Did the remedial lessons work? C. Did the remedial lessons work? _____ C. Did the remedial lessons work?
with the lesson. _____ No. of learners who have caught _____ No. of learners who have caught No. of learners who have caught up the _____ No. of learners who have caught
up the lesson. up the lesson. lesson. up the lesson.
D. No. of learners who continue to D. ___ No. of learners who continue to D. ___ No. of learners who continue to D. ___ No. of learners who continue to D. ___ No. of learners who continue to
require remediation require remediation require remediation require remediation require remediation
E. Which of my teaching strategies Strategies used that work well: Strategies used that work well: Strategies used that work well: Strategies used that work well:
worked well? Why did these work?
___ Group collaboration ___ Group collaboration ___ Group collaboration ___ Group collaboration

___ Games ___ Games ___ Games ___ Games

___ Poweerpoint presentation ___ Poweerpoint presentation ___ Poweerpoint presentation ___ Poweerpoint presentation

Answering preliminary Answering preliminary Answering preliminary Answering preliminary

 activities/exercises activities/exercises activities/exercises activities/exercises

___ Discussion ___ Discussion ___ Discussion ___ Discussion

___ Differentiated Instruction ___ Differentiated Instruction ___ Differentiated Instruction ___ Differentiated Instruction

___ Case Method ___ Case Method ___ Case Method ___ Case Method
___Role Playing /Drama ___Role Playing /Drama ___Role Playing /Drama ___Role Playing /Drama

___ Think-Pair-Share (TPS) ___ Think-Pair-Share (TPS) ___ Think-Pair-Share (TPS) ___ Think-Pair-Share (TPS)
___ Doscivery Method ___ Doscivery Method ___ Doscivery Method ___ Doscivery Method

___ Rereading of ___ Rereading of ___ Rereading of ___ Rereading of

Paragraphs/Poems/Stories ___ Lecture Paragraphs/Poems/Stories ___ Lecture Paragraphs/Poems/Stories ___ Lecture Paragraphs/Poems/Stories ___ Lecture
Method Method Method Method

Why? Why? Why? Why?

___ Complete Ims ___ Complete Ims ___ Complete Ims ___ Complete Ims

___ Availability of Materials ___ Availability of Materials ___ Availability of Materials ___ Availability of Materials

___ Pupil’s eagerness to learn ___ Pupil’s eagerness to learn ___ Pupil’s eagerness to learn ___ Pupil’s eagerness to learn

___ Group member’s cooperation in ___ Group member’s cooperation in ___ Group member’s cooperation in doing ___ Group member’s cooperation in
doing their tasks doing their tasks their tasks doing their tasks
F. What difficulties did I encounter ___ Bullying among learners ___ Bullying among learners ___ Bullying among learners ___ ___ Bullying among learners
which my principal or supervisor can ___ Equipment (AVR/LCD) ___ Equipment (AVR/LCD) Equipment (AVR/LCD) ___ Equipment (AVR/LCD)
help me solve?
___ Learner’s behavior/attitude ___ Learner’s behavior/attitude ___ Learner’s behavior/attitude ___ ___ Learner’s behavior/attitude
___ Science/Computer/Internet Lab ___ Science/Computer/Internet Lab Science/Computer/Internet Lab ___ Science/Computer/Internet Lab

___ Colorful Ims ___ Colorful Ims ___ Colorful Ims ___ ___ Colorful Ims
___ Additional Clerical Works ___ Additional Clerical Works Additional Clerical Works ___ Additional Clerical Works

___ Unavaailable Technology ___ Unavaailable Technology ___ Unavaailable Technology ___ ___ Unavaailable Technology
___ Reading Readiness ___ Reading Readiness Reading Readiness ___ Reading Readiness
G. What innovation or localized
materials did I use/discover which I
wish to share with other teachers?

Prepared by: Checked by: Noted:

NI-KHE G MARQUEZ MINERVA GRACE O. SILVESTRE EMELITA N. SORIANO, EdD

Teacher I, Subject Teacher Master Teacher I Principal IV

DAY 3 - DISCUSSIONS
 ATTACHMENT: (Developing Mastery)

Find the inverse of the following:

1. g(x) = x 3−2
2 x +1
2. f(x) =
3 x −4
Solution:

1. The equation of the function is y = x 3−2. Interchange the x and y variables: x = 3

y −2. Solve for y in terms of x:
3
x= y −2
x + 2 = y3

√3 x+ 2= y ⟹ y √3 x+ 2
y −2 is g ( x )=√ x+ 2.
3 −1 3
The inverse of g(x) =

2 x +1 2 x +1
2. The equation of the function is y = . Interchange the x and y variables: x= . Solve for y in terms of x:
3 x −4 3 x −4
2 x +1
x=
3 x −4
x(3y-4) = 2y + 1
3xy – 4x = 2y +1
3xy -2y = 4x + 1 (Place all terms with y on one side and those without y on the other side)
y (3x – 2) = 4x + 1
4 x +1
y=
3 x −2
−1 4 x+1
Therefore, the inverse of f(x) is f ( x )= .
3 x−2

APPLICATION
1. To convert from degrees Fahrenheit to Kelvin, the function is ,where t is the temperature in Fahrenheit (Kelvin is the SI unit of temperature). Find the inverse function converting
the temperature in Kelvin to degrees Fahrenheit.

Solution. The equation of the function is . To maintain k and t as the respective temperatures in Kelvin and Fahrenheit (and lessen confusion), let us not interchange the variables. We just
solve for t in terms of k: ,

Therefore the inverse is with temperature in Kelvin.

2. Give 3 examples of situations that can be represented as a one-to-one function and two examples of situations that are not one-to-one.

Sample answer:

 vehicles to plate numbers

 movie tickets to seat numbers
 presidents or prime ministers to countries
 mayors to cities or towns
3. Choose a situation or scenario that can be represented as one-to-one function and explain why it is important that the function in the scenario is one-to-one.
 A person must have only one tax Identification number (TIN) so that all the taxes he pays can be accurately recorded. If he has two TINs, the BIR might think that he did not pay all his taxes if his payments
are split between multiple TINs. If a single TIN has two persons associated to it, then it would not be possible to ascertain which person is paying the proper taxes and which is not.
EVALUATION

Find the inverse functions of the following one-to-one functions.

1
a) f(x) = x+ 4
2
b) f(x) = (x +3)3

3
c) f(x) =
x−4
x+ 3
d) f(x) =
x−3
2 x +1
e) f(x) =
4 x−1
Answer:

a) f −1(x) = 2x – 8

a) f −1(x) = 3 √x – 3
4 x +3
a) f −1(x) =
x
3 x +3
a) f −1(x) =
x −1
x +1
a) f −1(x) =
4 x−2