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General Mathematics

General Mathematics

Quarter 1 – Module 1 2 :

The Inverse of One-to-one

Functions

What I Need to Know

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

understand the inverse function. Particularly, this will provide you guide on how to

find the inverse of a one-to-one function. Enjoy as you immerse yourself in solving

for the inverse function intuitively or using a set of more established steps.

The module is composed of one lesson, namely:

• Lesson 1 – The Inverse of a One-to-one Function

After going through this module, you are expected to:

1. determine the inverse of a one-to-one function.
2. write a letter to a family member or peer about making amends on regretful

events which cannot be undone.

What I Know

Choose the letter of the best answer. Write the chosen letter on a separate sheet of

paper.

1. Which is not related to the word “inverse”?

a. undo

b. opposite

c. delete

d. interchange

1. What is the inverse of addition?

a. division

b. multiplication

c. subtraction

d. composition

1. What is the inverse of division?

a. addition

b. multiplication

c. subtraction

d. composition

1. What is the inverse of multiplication?

a. addition

b. division

c. subtraction

d. composition

1. What is the inverse of subtraction?

a. addition

b. division

c. multiplication

d. composition

1. Which function/s has/have inverse function?

a. one-to-one

b. many-to-one

c. both

d. none

1. What is the mathematical symbol for inverse of 𝑓(𝑥)?

a.

1

𝑓(𝑥)

b. 𝑓(𝑥)

− 1

c. ′𝑓(𝑥)

d. 𝑓

− 1

1. What is the inverse of 3x – 4?

a. 3x + 4

b. 4x – 3

c.

𝑥+ 3

4

d.

𝑥+ 4

3

1. What is the inverse of 𝑎

3

a. 3 a – 5

b. √𝑎 − 5

3

c. √

3

d.

𝑎

3

1. Which is not a property of an inverse function?

a. The inverse of 𝑓

− 1

(𝑥) is f(x).

b. 𝑓

− 1

) = 𝑥 for all x in the domain of f.

c. 𝑓

− 1

− 1

) = 𝑥 for all x in the domain of 𝑓

− 1

d. 𝑓

− 1

= 𝑥 for all x in the domain 𝑓

− 1

Lesson

The Inverse of One-to-One

Functions

Among the functions, only a one-to-one function has an inverse which is a function

also.

What’s In

So far, you have known different faces of functions in the previous lessons. Likewise,

you’ve categorized them already into groups of one-to-one and many-to-one

functions. Let’s have a quick review. In the first column, identify each of the following

as linear function (LF), quadratic function (QF) or rational function (RF). In the

second column, decide whether each is one-to-one or many-to-one function.

Function LF, QF or RF

One-to-one or many-to-

one

2

2

1

2

2 𝑥− 1

𝑥+ 5

Do you ever wonder if inverses of these functions are functions as well? That is, both

the original equation and its inverse are both functions. In this lesson, you will delve

into these functions with function inverses.

Notes to the Teacher

Solutions should be provided for exercises which will not be

successfully answered by the learners especially for “Additional

Activities” Part.

What’s New

I Can See Your Mind

Let’s have a mind game. Ready?

Think of a number. Multiply it by 2. Then, subtract 1 from it. Now, add 4 to the

difference. Lastly, give me your answer and I’ll tell the number you are thinking of.

Can you tell me how I will know the original number you have chosen by giving me

the final answer?

The key lies in the command “undo”. Familiar with it? Yes, this game follows the

same principle as with the “undo” button we click when we are preparing documents

using our laptops, cellphones or the likes. When you want to bring back how the

document looks like a while ago, you keep clicking this button and the document

gradually goes back to its previous layout. It keeps deleting the changes you do to

the document one by one from the most recent to the earliest change you made.

Meanwhile, what you did with your chosen number is you multiplied it by 2 and then

added 3 to it. Why 3? Because you subtracted 1 and then added 4 to the number

which is the same as adding 3 to it. Going back to the principle of “undo”, this is how

I guessed your original number by telling me your final answer.

Commands Undo

Step 3. Add 3 to it. (2x + 3) Step 1. Subtract 3 from your answer y. (y – 3)

Step 2. Multiply it by 2. (2x) Step 2. Divide it by 2.

𝑦− 3

2

Step 1. Think of a number. (x) Answer will be the number you are thinking. (x)

By that way, I have seen your mind. Create a new set of commands. It’s now your

turn to try it with your family members or peer. Experience their oohs and aahs!

What is It

Inverse Function Defined

The inverse of a function is a function with domain B and range A given that the

original function has domain A and range B.

This inverse function of function f is denoted by f

• 1 . It is defined by the equation

− 1

= 𝑥 if and only if 𝑓(𝑥) = 𝑦 for any y in range B. Since both are functions, then

a function has to be one-to-one for its inverse to be a function at the same time. If it

is a many-to-one function, its inverse is one-to-many which is not a function.

Example 3

Find the inverse of the rational function ℎ

4 𝑥+ 8

𝑥− 3

Solution:

4 𝑥+ 8

𝑥− 3

(change g(x) to y)

4 𝑦+ 8

𝑦− 3

(interchange x and y)

𝑥𝑦 − 3 𝑥 = 4 𝑦 + 8 (solve for y, MPE)

𝑥𝑦 − 4 𝑦 = 3 𝑥 + 8 (solve for y, by APE)

𝑦(𝑥 − 4 ) = 3 𝑥 + 8 (solve for y, by factoring)

3 𝑥+ 8

𝑥− 4

(solve for y, by MPE)

− 1

3 𝑥+ 8

𝑥− 4

(the inverse function)

What’s More

Activity 12.

Intuitively, give the inverse function of each of the following.

𝑥

4

3 𝑥+ 5

8

Activity 12.

If it exists, solve for the inverse function of each of the following.

12 𝑥− 1

7

9 𝑥

4

1

3

9

3

2

2 𝑥+ 17

3 𝑥+ 1

𝑥+ 10

9 𝑥− 1

What I Have Learned

Answer the following questions.

1. What is an inverse function?

1. What is the symbol of an inverse function?
2. Do all kinds of functions have inverse function?
3. How do you solve for the inverse of a one-to-one function?

What I Can Do

In real life, can we undo events? Have you experienced any conflict with your family

or peer on concerns like showing respect, being honest and trustworthy or being

helpful and cooperative? What do you do to make amends? This time try writing a

letter to a family member or peer expressing your regret over an event. Pour out your

heart and feel light after then.

To make sure you’ll put smile on their faces, try scoring your letter using the rubric

below:

Criteria 4 3 2 1

Content Focus on actions to

take to resolve the

situation; sincere and

polite tone; admit one’s

fault; with follow up

Involves only

three of the

four

characteristics

cited at the left

Involves only

two of the four

characteristics

cited

Involves only

one of the four

characteristics

cited

Grammar

and

mechanics

Sentences are clear; use

commas and other

punctuations properly;

no lengthy narration in

every sentence;

sentences are arranged

properly

Involves only

three of the

four

characteristics

cited at the left

Involves only

two of the four

characteristics

cited

Involves only

one of the four

characteristics

cited

If you scored your letter 7 or 8, proceed giving your letter wholeheartedly. If the score

you give is 6 or below, consider revising it before giving it to your loved one. This is a

rare moment, make it count.

Assessment

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

separate sheet of paper.

1. Which relates to “inverse”?

a. redo

b. opposite

c. delete

d. interchange

1. What is the inverse of 𝑓(𝑥) = 𝑥 + 36?

a. The inverse of 𝑓

− 1

(𝑥) is 𝑓(𝑥).

b. 𝑓

− 1

= 𝑥 for all x in the domain of 𝑓.

c. 𝑓

− 1

− 1

(𝑥)) = 𝑥 for all x in the domain of 𝑓

− 1

d. 𝑓(𝑓

− 1

(𝑥)) = 𝑥 for all x in the domain of 𝑓

− 1

1. What is the inverse of − 6 𝑥 − 5?

a.

𝑥+ 5

− 6

b. −

𝑥

6

c. 6 𝑥 + 5

d. 5 𝑥 + 6

1. What is the inverse of

3

a. √

3

b. √𝑐 + 1

3

c. √

3

d. 1 − √𝑐 + 1

3

1. What is the inverse of 𝑔(𝑥) = 9 𝑥 + 20?

a. 𝑔

′

𝑥+ 20

− 9

b. 𝑔

′

c. 𝑔

− 1

𝑥− 20

9

d. 𝑔

− 1

1. What is the inverse function of 𝑓

𝑑− 12

2 𝑑+ 1

a. 𝑓

′

𝑑+ 12

− 2 𝑑− 1

b. 𝑓

′

−𝑑+ 12

− 2 𝑑− 1

c. 𝑓

′

𝑑− 12

2 𝑑− 1

d. 𝑓

′

−𝑑− 12

2 𝑑− 1

1. Which is not involved in the process of finding the inverse of a function?

a. Write the function in the form 𝑦 = 𝑓(𝑥).

b. Interchange the x and y variables.

c. Write in the function in the simplest form.

d. Solve for y in terms of x.

Additional Activities

Show that 𝑓

has no inverse function.

The following rubric will be used in rating your work:

Answer Key

Assessment

D 1.

C 2.

B 3.

B 4.

A 5.

A 6.

D 7.

D 8.

C 9.

C 10.

A 11.

B 12.

C 13.

D 14.

C 15.

What's More

Activity 12.

1. 𝑓

− 1

(𝑥) = 𝑥 − 2

1. 𝑔

− 1

(𝑥) =

𝑥+ 1

12

1. ℎ

− 1

( 𝑥

) = − 4 𝑥

1. 𝑓

− 1

( 𝑥

) = 𝑥

1. 𝑔

− 1

( 𝑥

)

8 𝑥− 5

3

Activity 12.

1. 𝑓

− 1

(𝑥) =

𝑥+ 18

25

1. 𝑔

− 1

(𝑥) =

7 𝑥+ 1

12

1. ℎ

− 1

( 𝑥

)

− 4 ቀ𝑥+

1

3

ቁ

9

𝑜𝑟 ℎ

− 1

( 𝑥

) = −

4 𝑥

9

4

27

1. 𝑓

− 1

( 𝑥

)

√

𝑥

9

1. 𝑓

− 1

(𝑎) = √

𝑎 − 8

3

1. It has no inverse.

Initially, the given

is not a one-to-one

function. Or, by

solving for the

inverse, 𝑦 =

± √

𝑥 + 23 − 4.

There are y-values

each of which is

paired to two x-

values.

1. It has no inverse.

Initially, the given

is not a one-to-one

function. Or, by

solving for the

inverse, 𝑦 =

± √

𝑥 + 16 − 2.

There are

ordinates each of

which is paired to

two abscissas.

1. ℎ

− 1

(𝑥) =

2 𝑥+ 17

3 𝑥+ 1

1. ℎ

− 1

( 𝑐

)

𝑥

2

− 2

2

1. 𝑓

− 1

( 𝑥

)

𝑥+ 10

9 𝑥− 1

What I Know

C 1.

C 2.

B 3.

B 4.

A 5.

A 6.

D 7.

D 8.

B 9.

C 10.

A 11.

D 12.

C 13.

B 14.

C 15.

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