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General

Mathematics

Quarter 1 – Module 4:

Inverse Functions

General Mathematics – Grade 11

Self-Learning Module (SLM)

Quarter 1 – Module 4: Inverse Functions

First Edition, 2020

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Development Team of the Module

Writers: Ana Cristy P. Mercado, Ian Caesar E. Frondoza, Gelyn L. Rivera, Javy B. Nuñez,

Fritz P. Lejesta

Editors: Ian Caesar E. Frondoza

Reviewers: Reynaldo C. Tagala

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Judith B. Alba – EPS, ADM

Reynaldo C. Tagala – EPS, Mathematics

Introductory Message

For the facilitator:

Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on

Inverse Functions!

This module was collaboratively designed, developed and reviewed by educators both

from public and private institutions to assist you, the teacher or facilitator in helping

the learners meet the standards set by the K to 12 Curriculum while overcoming

their personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent

learning activities at their own pace and time. Furthermore, this also aims to help

learners acquire the needed 21st century skills while taking into consideration their

needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of

the module:

As a facilitator you are expected to orient the learners on how to use this module.

You also need to keep track of the learners' progress while allowing them to manage

their own learning. Furthermore, you are expected to encourage and assist the

learners as they do the tasks included in the module.

Notes to the Teacher

This contains helpful tips or strategies that

will help you in guiding the learners.

For the learner:

Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on

Inverse Functions!

The hand is one of the most symbolized part of the human body. It is often used to

depict skill, action and purpose. Through our hands we may learn, create and

accomplish. Hence, the hand in this learning resource signifies that you as a learner

is capable and empowered to successfully achieve the relevant competencies and

skills at your own pace and time. Your academic success lies in your own hands!

This module was designed to provide you with fun and meaningful opportunities for

guided and independent learning at your own pace and time. You will be enabled to

process the contents of the learning resource while being an active learner.

This module has the following parts and corresponding icons:

What I Need to Know

This will give you an idea of the skills or

competencies you are expected to learn in the

module.

What I Know

This part includes an activity that aims to

check what you already know about the

lesson to take. If you get all the answers

correct (100%), you may decide to skip this

module.

What’s In

This is a brief drill or review to help you link

the current lesson with the previous one.

What’s New

In this portion, the new lesson will be

introduced to you in various ways such as a

story, a song, a poem, a problem opener, an

activity or a situation.

What is It

This section provides a brief discussion of the

lesson. This aims to help you discover and

understand new concepts and skills.

What’s More

This comprises activities for independent

practice to solidify your understanding and

skills of the topic. You may check the

answers to the exercises using the Answer

Key at the end of the module.

What I Have Learned

This includes questions or blank

sentence/paragraph to be filled in to process

what you learned from the lesson.

What I Can Do

This section provides an activity which will

help you transfer your new knowledge or skill

into real life situations or concerns.

What I Need to Know

This module was designed and written with you in mind. It is here to help you master

the Inverse of a One-to-one Function. The scope of this module permits it to be used

in many different learning situations. The language used recognizes the diverse

vocabulary level of students. The lessons are arranged to follow the standard

sequence of the course. But the order in which you read them can be changed to

correspond with the textbook you are now using.

The module is outlined in this manner:

• Lesson 1: Representing real-life situations using one-to-one functions
• Lesson 2: Inverse of a one-to-one function.
• Lesson 3: Representing an inverse function through its:

(a) table of values, and

(b) graph.

• Lesson 4: Domain and Range of an inverse function.
• Lesson 5: Problems involving inverse functions.

After going through this module, you are expected 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. illustrate the inverse of a function using table of values and graph

(M11GM-Id- 3 );

4. determine the domain and range of inverse functions (M11GM-Id-4); and
5. solve problems involving inverse functions (M11GM-Ie-2).

What I Know

Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a

separate sheet of paper.

1. Which of the following ordered pair represents one to one functions.

A. (0,2)(1,2)(2,2)

B. (-2,0)(-2,1)(-2,-2)

C. (0,1)(1,2)(2,3)

D. (0,0)(1,3)(2,6)

2. One to one functions crosses a horizontal line ______ times.

A. 0

B. 1

C. 2

D. infinitely many

3. All of the following are one to one functions, EXCEPT:

A. 𝑓(𝑥) = 𝑥

B. 𝑓(𝑥) = 1

C. 𝑓(𝑥) = 𝑥 − 1

D. 𝑓(𝑥) = 𝑥 + 1

4. Complete the statement: A function is one to one if:

A. exactly one domain corresponds to exactly one range.

B. there is two domains in one range.

C. in every domain there corresponds two ranges.

D. many domain and many range.

5. Complete the statement: (-1,2) (1,2) (2,2) is:

A. one to one

B. one to many

C. many to one

D. many to many

6. The inverse of an inverse 𝑓

− 1

(𝑥) is ________.

A. 𝑥

B. 1

C. 𝑓

− 1

D. 𝑓(𝑥)

− 1

(𝑓(𝑥)) = _______ for all 𝑥 in the domain of 𝑓.

A. y

B. x

C. 1

D. 0

8. A function with an inverse is described to be _________________.

A. one-to-many

B. one-to-one

C. many-to-one

D. many-to-many

16. Given a one-to-one function f(x) and its inverse f^(-1)(x). Then the following

are true except one:

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 range of

− 1

f

D. 𝑓

− 1

( f ( x )) = x for all x in the domain of f

17. What is the result if a function that is not one-to-one is inverted?

A. not a function

B. a function

C. a relation

D. not a relation

18. Find the domain of this set of data. { (1,2), (3,4), (5,6), (7,8) }

A. { 2, 4, 6, 8}

B. { 1, 3, 5, 7 }

C. { 1, 2, 3, 4 }

D. { 5, 6, 7, 8 }

19. The range f of an inverse function is the __________ of the original function.

A. domain

B. range

C. Inverse

D. Function

20. A function with an inverse is described to be _________________.

A. one-to-many

B. one-to-one

C. many-to-one

D. many-to-many

Lesson

1

Representing Real-life

Situations Using One-to-one

Functions

Have you ever asked yourself how your height and weight predict your BMI

(Body Mass Index)? How does the power consumption affects the electric bill? How

does the number of studying hours affect your grade? This module will discuss

further how to determine one to one functions and its application in real life

situations.

What’s In

Group the following objects in such a way that the way that they common

properties/characteristics.

triangle guava circle

pineapple crayons eraser

apple rectangle durian

pencil book yellow pad

oval banana square

Shapes Fruits School Supplies

Form some ordered pairs using the format:

a. Column 1 ____________________________________________________

b. Column 2 ____________________________________________________

c. Column 3 ____________________________________________________

Questions:

1. How many objects can be found in each column?
2. How did you classify the objects?
3. Based on the coordinates you have formulated, is there a repetition of the

first coordinates? What about the second coordinates?

Example 1: Suppose the Kidzoona charge in an hour play is Php 300. Your baby

brother wants to play in the said amusement.

Questions:

1. How much will you pay if your baby brother will play 1 hour? 2 hours? 3

hours? How about 5 hours?

2. Based on your answer in item 1, write an ordered pair in the form (time,

amount).

3. Make a rule/ function that represents the given relation.
4. Is the given relation a one to one function?
5. How are time consume in Kidzoona related to charge to be paid?

Solutions:

1. If the brother will play 1 hour the sister will pay Php 300, while if he will

play 2 hours his sister will pay Php 600, moreover if he will play for 3 hours

they will pay Php 900.

2. Ordered pairs are: (1,300) (2,600) (3,900).
3. 𝑓(𝑥) = 300 𝑥, thus 𝑥 represents the time spent in Kidzoona while f(x)

represents the total charge to be paid.

4. The relation is a one to one function since there exactly one element of f(x)

in every element of x.

5. The total amount to be paid depends on the time consumed in Kidzoona.

Example 2 : Suppose you want to call your father by telephone. The charge of a pay

phone call is Php 10 pesos for the first 2 minutes and an additional Php 3 for every

additional minute or a half of it.

Questions:

1. How much will you pay if you call your father for 1 minute? 2 minutes?

How about 4 minutes?

2. Represents table of value of the given relation based on your answer in item

3. Is the relation a one to one function? Why or why not?

Solutions:

1. If I will call my father for 1 minute I will pay Php 10, if 2 minutes Php 10

and if 4 minutes Php 16.

Number of minutes

of phone call (x)

Total amount (f(x)) Php 10 Php 10 Php 16

3. The relation is not a one to one function since there is only one element of

f(x) in every element of x. Thus, it represents a many to one function.

A relation has four types:

• One to one
• Many to one
• One to many
• Many to many; thus only one to one relation and many to one relation

represents a function.

Lesson

2

Inverse of a One-to-one

Function

A function and its inverse can be described as the "DO" and the "UNDO"

functions. A function takes a starting value, performs some operation on this value,

and creates an output answer. The inverse function takes the output answer,

performs some operation on it, and arrives back at the original function's starting

value.

Inverting 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.

To find the inverse of a one-to-one function,

(a) write the function in the form 𝑦 = 𝑓(𝑥);

(b) interchange the x and y variables;

(c) solve for y in terms of x. This is because we are interchanging the input

and output values of a function.

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

Solution:

a. write the function in the form 𝑦 = 𝑓(𝑥). → 𝑦 = 3 𝑥 + 1

b. interchange the x and y variables. → 𝑥 = 3 𝑦 + 1

c. solve for y in terms of x. → 𝑥 = 3 𝑦 + 1

𝑥− 1

3

=

3 𝑦

3

𝑥− 1

3

= 𝑦

Therefore, the inverse of 𝒇(𝒙) = 𝟑𝒙 + 𝟏 is 𝒇

−𝟏

𝒙−𝟏

𝟑

Definition

Let 𝑓 be a one-to-one function with domain 𝐴 and range 𝐵. Then the 𝑖𝑛𝑣𝑒𝑟𝑠𝑒

𝑜f 𝑓, denoted by 𝑓

− 1

, is a function with domain 𝐵 and range 𝐴 defined by 𝑓

− 1

if and only if 𝑓(𝑥) = 𝑦 for any 𝑦 in 𝐵.

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

As shown earlier, 'inverting' the x- and y-values of a function results in a

function if and only if the original function is one-to-one.

Example 2: Find the inverse of 𝑔

3

Solution:

3

3

3

3

3

3

3

𝟑

−𝟏

Therefore, the inverse of 𝑔(𝑥) = 𝑥

3

− 2 is 𝒈

−𝟏

𝟑

Example 3: Find the inverse of 𝑓

2

• 4 𝑥 − 2 if it exists.

Solution:

2

2

2

2

• 4 𝑦 + 4 (Complete the square)

2

2

−𝟏

The equation ± √

𝑥 + 6 − 2 = 𝑦 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.

Example 4: Find the inverse of 𝑓(𝑥) = | 4 𝑥|, if it exists.

Solution: Recall that the graph of 𝑦 = | 3 𝑥| is shaped like a

“V” whose vertex is located at the origin. This

function fails the horizontal line test and therefore

has no inverse.

Alternate Solution: We can also show that 𝑓

− 1

does not exist by

showing that 𝑓 is not one-to-one. Note that

𝑓( 1 ) = 𝑓(− 1 ) = 4. Since the x-values 1 and - 1 are

paired to the same y-value, then 𝑓 is not one-to-

one and it cannot have an inverse.

If we apply the procedure in solving the inverse of a one-to-one

function:

2

(Recall the definition |𝑥| = √𝑥

2

)

2

2

𝑥

2

4

= 𝑦

2

±

√

𝑥

2

4

= 𝑦

Here, 𝑥 = 2 will correspond to 𝑦 = 1 and 𝑦 = − 1 , so 𝑦 = ±

𝑥

2

4

is not a

function. Therefore, 𝑓

has no inverse function.

Lesson

3

Table and Graph of an

Inverse Function

If function passes the horizontal line test, it is an indication that the said

function is one-to-one. Subsequently, its inverse also is a function.

Since the domain of the function becomes the range of its inverse, and the

range of the function becomes the domain of its inverse, in problems dealing with a

function and its inverse, when determining the inverse’ ordered pairs, we simply

reverse the domain and range of the original function.

Example 1

Find the inverse of the function 𝑓(𝑥) = 2 𝑥 + 1 ; present the graph of the original

function and its inverse.

x - 1 0 1 2

y

The accomplished table of ordered pairs of the original function 𝑓

= 2 𝑥 + 1 and its

inverse is presented below.

− 1

𝑥− 1

2

x - 1 0 1 2

y - 1 1 3 5

The graphs of the given function and its inverse are reflections of each other along

the symmetry line 𝑥 = 𝑦.

x - 1 1 3 5

y - 1 0 1 2

In order to accomplish a table of values

such this, first, you must set values of

your x.

Then, to obtain corresponding

values of y, substitute each values of x in

the function.

We have the first value of x that is - 1,

y=2x+

y=2(-1) +

y=-2+

y=-1.

Lesson

4

Domain and Range of an

Inverse Function

Since the domain of the function becomes the range of its inverse, and the

range of the function becomes the domain of its inverse, in problems dealing with a

function and its inverse, when determining the inverse’ ordered pairs, we simply

reverse the domain and range of the original function.

EXAMPLE 1.

Graph y = 𝑓

− 1

( x ) if the graph of y = f ( x ) =

2 x + 1 restricted in the domain

{x |− 2 ≤ x ≤ 1_._ 5 } is given below. What is

the range of f ( x )? What is the domain and

range of its inverse?

Solution.

Take the reflection of the restricted graph of y = 2 x + 1 across the line y = x.

The range of the original function can be determined by inspection of the graph.

The range is {y ∈ R | − 3 ≤ y ≤ 4 }.

Domain [ − 2 , 1_._ 5] [ − 3 , 4]

Range [ − 3 , 4] [ − 2 , 1_._ 5]

In summary:

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. Is this true for all one-to-

one functions and their inverses?

EXAMPLE 2.

Find the domain and range the inverse of f ( x ) =

1

𝑥

whose graph is shown below.

Solution.

Applying the horizontal line test we

verify that the function is one-to-one.

Since the graph of f ( x ) =

1

𝑥

is 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 ).

In summary:

The domain and range of the inverse function can be determined by inspection of

the graph:

Domain of 𝑓

− 1

( x ) = [ − 3 , 4]

Range of 𝑓

− 1

( x ) = [ − 2 , 1_._ 5]

f ( x ) 𝑓

− 1

( x )

Domain [ - 2, 1, 5] [ - 3, 4]

Range [ - 3, 4] [ - 2, 1, 5]