General Mathematics Quarter 1 Module 14: Domain and Range of Inverse Functions

General Mathematics Quarter 1 – Module 14: Domain and Range of Inverse Functions

Senior High School

[illustration: group of students and a teacher studying outdoors and at a table, with a globe and a lamp]

General Mathematics Alternative Delivery Mode Quarter 1 – Module 14: Domain and Range of Inverse Functions First Edition, 2020

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer: Dennis E. Ibarrola

Editors: Elizabeth B. Dizon, Anicia J. Villaruel, and Roy O. Natividad

Reviewers: Fritz A. Caturay, Necitas F. Constante, Dexter M. Valle, Jerome A. Chavez, Angelo S. Villanueva, Charito N. Laguador and Moahna Aura M. Mancenido

Illustrator: Dianne C. Jupiter and Michael A. Alonzo

Layout Artist: Noel Rey T. Estuita

Management Team: Francis Cesar B. Bringas
Job S. Zape, Jr.
Ramonito Elumbaring
Reicon C. Condes
Elaine T. Balaogan
Fe M. Ong-ongowan
Hermogenes M. Panganiban
Phillip B. Gallendez
Josephine T. Natividad
Anicia J. Villaruel
Dexter M. Valle

Printed in the Philippines by ________________________

Department of Education – Region IV-A CALABARZON

Office Address: Gate 2 Karangalan Village, Barangay San Isidro, Cainta, Rizal 1800 Telefax: 02-8682-5773/8684-4914/8647-7487 E-mail Address: region4a@deped.gov.ph

General Mathematics Quarter 1 – Module 14: Domain and Range of Inverse Functions

Senior High School

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

In this learning module, you will know more about the domain and range, and how to determine the domain and range of an inverse function. This module was designed and written with you in mind. It is here to help you easily master the procedure in finding the domain and range of an inverse function.

After going through this module, you are expected to:

1. Define domain and range.
2. Find the domain and range of a given inverse function.
3. Represent the domain and range using set builder notation.

What I Know

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

1. What do you call to the set of all allowable values of the independent variable?
   a. Range
   b. Domain
   c. Real Numbers
   d. Inverse Function

   
   

2. Which of the following is not allowed as the value of the independent variable if the function is a fraction?
   a. zero
   b. negative number
   c. decimal number
   d. irrational number

   
   

3. What is the domain and range of the function (𝑥𝑥) = 𝑥𝑥 − 5 ? [transcriber’s note: the function name is not visible in the page image; the printed expression is reproduced as it appears]
   a. The domain is all real numbers except -5 and the range is all real numbers except 0.
   b. The domain is all real numbers and the range is all real numbers except 0.
   c. The domain is all real numbers except -5 and the range is all real numbers.
   d. The domain and range are all real numbers.

   
   

4. What is the inverse of (𝑥𝑥) = 3𝑥𝑥 + 6 ? [transcriber’s note: the function name is not visible in the page image; the printed expression is reproduced as it appears]
   a. 𝑓𝑓⁻¹(𝑥𝑥) = (𝑥𝑥+6)/3
   b. 𝑓𝑓⁻¹(𝑥𝑥) = 3/(𝑥𝑥−6)
   c. 𝑓𝑓⁻¹(𝑥𝑥) = (𝑥𝑥−6)/3
   d. 𝑓𝑓⁻¹(𝑥𝑥) = 3/(𝑥𝑥+6)

   
   

5. Which of the following pairs of functions is NOT the inverse of each other?
   a. 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 + 5 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 5
   b. 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔(𝑥𝑥) = 𝑥𝑥/3
   c. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 − 3 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 3
   d. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 2 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2

   
   

For numbers 6-10, consider the function 𝑓𝑓(𝑥𝑥) = 3/(𝑥𝑥+1).

6. What is the domain of the function?
   a. {𝑥𝑥 Є 𝑅𝑅}
   b. {𝑥𝑥 ≠ 0}
   c. {𝑥𝑥 ≠ −1}
   d. {𝑥𝑥 ≠ 1}

[transcriber’s note: the numbers after item 6 are printed as 1 to 4 on the page; they are transcribed as printed]

1. What is the Range of the function?
   a. {𝑦𝑦 ≠ 0}
   b. {𝑦𝑦 ≠ 1}
   c. {𝑦𝑦 ≠ 3}
   d. {𝑦𝑦 ≠ −1}

   
   

2. What is the inverse of the function?
   a. 𝑓𝑓⁻¹(𝑥𝑥) = (3+𝑥𝑥)/𝑥𝑥
   b. 𝑓𝑓⁻¹(𝑥𝑥) = (3−𝑥𝑥)/𝑥𝑥
   c. 𝑓𝑓⁻¹(𝑥𝑥) = 𝑥𝑥/(3+𝑥𝑥)
   d. 𝑓𝑓⁻¹(𝑥𝑥) = 𝑥𝑥/(3−𝑥𝑥)

   
   

3. What is the domain of 𝑓𝑓⁻¹?
   a. {𝑥𝑥 ≠ 0}
   b. {𝑥𝑥 ≠ 3}
   c. {𝑥𝑥 ≠ −3}
   d. {𝑥𝑥 ≠ −1}

   
   

4. What is the range of 𝑓𝑓⁻¹?
   a. {𝑦𝑦 ≠ 0}
   b. {𝑦𝑦 ≠ 𝑦𝑦}
   c. {𝑦𝑦 ≠ 1}
   d. {𝑦𝑦 ≠ −1}

   
   

For numbers 11-15, consider the function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥² + 2.

11. What is the domain of the function?
    a. {𝑥𝑥 > 2}
    b. {𝑥𝑥 Є ≠ 2}
    c. {𝑥𝑥 > 0}
    d. {𝑥𝑥 Є 𝑅𝑅}

    
    

12. What is the range of the function?
    a. {𝑦𝑦 ≥ 2}
    b. {𝑦𝑦 > 2}
    c. {𝑦𝑦 < 2}
    d. {𝑦𝑦 > 0}

    
    

13. What is the inverse of the function?
    a. 𝑓𝑓⁻¹(𝑥𝑥) = 𝑥𝑥² − 2
    b. 𝑓𝑓⁻¹(𝑥𝑥) = 2 + 𝑥𝑥²
    c. 𝑓𝑓⁻¹(𝑥𝑥) = √(𝑥𝑥 − 2)
    d. 𝑓𝑓⁻¹(𝑥𝑥) = √(𝑥𝑥 + 2)

    
    

14. What is the domain of 𝑓𝑓⁻¹?
    a. {𝑥𝑥 ≥ −2}
    b. {𝑥𝑥 ≥ 2}
    c. {𝑥𝑥 < −2}
    d. {𝑥𝑥 < 2}

    
    

15. What is the range of 𝑓𝑓⁻¹?
    a. {𝑦𝑦 Є 𝑅𝑅}
    b. {𝑦𝑦 Є > 2}
    c. {𝑦𝑦 < 2}
    d. {𝑦𝑦 > −2}

    
    

Lesson 1

Finds the Domain and Range of an Inverse Function

Start Lesson 1 of this module by assessing your knowledge of the basic skills in finding the inverse of a function. This knowledge and skill will help you understand easily on how to find the domain and range of an inverse function. Seek the assistance of your teacher if you encounter any difficulty. This topic is about finding the domain and range of an inverse function.

What’s In

[illustration: hand holding a puzzle piece]

Recall that a function has an inverse if and only if it is one-to-one and every one-to-one function has a unique inverse function.

Below are the steps in solving for the inverse of a 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; d. Replace y by f-1(x); e. Verify if f and f-1 are inverse functions.

Example 1: Find the inverse of 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 − 8.

Solution: The equation of a function is 𝑦𝑦 = 3𝑥𝑥 − 8. Interchanging the x and y variables, we get 𝑥𝑥 = 3𝑦𝑦 − 8.

Solving y for x: 3𝑦𝑦 = 𝑥𝑥 + 8 𝑦𝑦 = (𝑥𝑥+8)/3

Therefore, the inverse of 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 − 8 is 𝑓𝑓⁻¹(𝑥𝑥) = (𝑥𝑥+8)/3.

To verify if f and f-1 are inverse functions:

𝑓𝑓[𝑓𝑓⁻¹(𝑥𝑥)] = 3((𝑥𝑥+8)/3) − 8 = 𝑥𝑥+8 = 𝑥𝑥 𝑓𝑓⁻¹[𝑓𝑓(𝑥𝑥)] = (3𝑥𝑥−8+8)/3 = 3𝑥𝑥/3 = 𝑥𝑥

[transcriber’s note: the two verification columns are printed side by side on the page; each is shown on one line here, and the printed line “= x+8” is followed by “=x” as shown]

Therefore, f-1 is the inverse of f.

I can’t transcribe pages 9 to 16 because they aren’t in this message. The only page content I can see is the eight pages labeled 1 to 8, which carry module pages 5 to 12. Those pages were already transcribed separately, so I haven’t repeated them.

Please upload or attach the rendered images of pages 9 to 16 of the PDF. I’ll then transcribe them with page markers 9 to 16, following the same structure and rules as before.

[transcriber’s note: this page continues an activity from the previous page]

4. f(x) = x² + 2

f⁻¹ =	Solution:
Domain
Range

5. f(x) = √(1 + x)

f⁻¹ =	Solution:
Domain
Range

What I Have Learned

Think It Over And Complete Me!

A. Complete The Paragraph

Remember that an inverse function is a _____ function. Whereas, the _____ of the inverse function is the range of the one-to-one function and the _____ of the inverse function is the domain of the one-to-one function.

To find the domain and range of an inverse function, go back to the _____ function and then _____ the domain and range of the original function.

B. How is the skill in operating fractions and radicals relevant in determining the domain and range of the inverse function? Explain.

---

---

---

---

---

---

---

C. You have understood that inverse function is a function that reverses another function. In life, if it so happens that you have done some mistakes, you can only correct it and not reverse it. But if you would be given a chance to reverse one thing in your life, what would it be and why?

---

---

---

---

---

---

---

What I Can Do

EXPLORE DEEPER AND THINK WISELY

Now that you have deeper understanding of the topic, you are ready to solve the problems below.

1. Temperatures are normally measured in degrees Celsius or degrees Fahrenheit. A temperature reading expressed in degrees Celsius can be converted to degrees Fahrenheit, and vice versa.

a. Determine a function F that expresses a given temperature in degrees Fahrenheit to degrees Celsius. Solution:

b. Determine a function C that expresses a given temperature in degrees Celsius to degrees Fahrenheit. Solution:

c. Verify if the functions F and C are inverse Functions. Solution:

d. Determine the domain and range of the functions and its inverse.

2. The formula S= (n-2) 180 gives the sum of the measures of the angles of n-sided polygon where n is the input and S is the output.

a. Solve the formula for n so that S becomes the input and n becomes the output. Solution:

b. Write the formula in (a) as the inverse function of f(x) = (x-2) 180. Solution:

c. Verify is the two functions are inverse functions. Solution:

d. Determine the domain and range of the function and its inverse.

Rubrics:

Score	For letters a and b	For letter c	For letter d
4	The function or formula was determined or formulated with properly shown procedures.	The functions were verified as inverse functions with completely shown procedures.	The domain and range of the function and its inverse were correctly determined and properly written.
3	The function or formula was determined or formulated with partially shown procedures.	The functions were verified as inverse functions with partially shown procedures.	The domain and range of the function and its inverse were correctly determined but it was improperly written.

[transcriber’s note: table continued from the previous page]

Score	For letters a and b	For letter c	For letter d
2	The function or formula was not determined or formulated and other alternative procedures was shown.	The functions were not verified as inverse functions and other alternative procedures was shown.	The domain was correct but the range is incorrect or vice versa.
1	The function or formula was not determined or formulated without any procedure or solution.	The functions were not verified as inverse functions without any procedure.	The domain and range was incorrectly determined and improperly written.

Assessment

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

1. What do you call to the resulting y-values after we have substituted in the possible x – values?
   a. Range
   b. Domain
   c. Real Numbers
   d. Inverse Function

   
   

2. Which of the following is not allowed as the value of the independent variable under the square root sign?
   a. zero
   b. negative number
   c. decimal number
   d. fraction

   
   

3. What is the domain and range of the function (x) = 5x + 2 ?
   a. The domain is all real numbers except 2 and the range is all real numbers except 0.
   b. The domain is all real numbers and the range is all real numbers except 0.
   c. The domain is all real numbers except 2 and the range is all real numbers.
   d. The domain and range are all real numbers.

   
   

4. What is the inverse of (x) = 9x + 5 ?
   a. f⁻¹(x) = (x−5)/9
   b. f⁻¹(x) = 9/(x−5)
   c. f⁻¹(x) = (x+5)/9
   d. f⁻¹(x) = 9/(x+5)

   
   

b. f⁻¹(x) = 9/(x−5) d. f⁻¹(x) = 9/(x+5)

5. Which of the following pair of functions is NOT the inverse of each other?
   a. f(x) = 5x and g(x) = x/5
   b. f(x) = 2 − 3x and g(x) = (2−x)/3
   c. f(x) = x and g(x) = 1/x
   d. f(x) = x² and g(x) = √x

For numbers 6-10, consider the function f(x) = 3/(x−2).

6. What is the domain of the function?
   a. {x ≠ 3}
   b. {x ≠ 0}
   c. {x ≠ 2}
   d. {x ≠ −2}

   
   

7. What is the Range of the function?
   a. {y ≠ −2}
   b. {y > 0}
   c. {y ≠ 3}
   d. {y ≠ 0}

   
   

8. What is the inverse of the function?
   a. f⁻¹(x) = (2x−3)/x
   b. f⁻¹(x) = (2x+3)/x
   c. f⁻¹(x) = x/(2x−3)
   d. f⁻¹(x) = x/(2x+3)

   
   

9. What is the domain of f⁻¹?
   a. {x ≠ 0}
   b. {x ≠ 2}
   c. {x ≠ −2}
   d. {x ≠ 3}

   
   

10. What is the range of f⁻¹?
    a. {y ≠ 3}
    b. {y ≠ −3}
    c. {y ≠ −2}
    d. {y ≠ 2}

    
    

For numbers 11-15, consider the function (x) = √(x − 1) .

11. What is the domain of the function?

a. {x ≥ −1} b. {x ≥ 1} c. {x ≥ 0} d. {x < 1}

12. What is the range of the function?
    a. {y < 1}
    b. {y > 0}
    c. {y ∈ R}
    d. {y > 1}

    
    

13. What is the inverse of the function?
    a. f⁻¹(x) = x² − 1
    b. f⁻¹(x) = x² + 1
    c. f⁻¹(x) = 1/x²
    d. f⁻¹(x) = x²

    
    

14. What is the domain of f⁻¹?
    a. {x > 1}
    b. {x ∈ R}
    c. {x < 1}
    d. {x > 0}

    
    

15. What is the range of f⁻¹?
    a. {y ≥ 1}
    b. {y ≥ −1}
    c. {y < 1}
    d. {y ≥ 1}.

    
    

Additional Activities

Give Me More Companions

In this section, you are going to think deeper and test further your understanding of domain and range of inverse function. Ask someone who can help you to find the correct solutions and answer.

Tom and Jerry are school mates and they are playing a number-guessing game. Tom asks Jerry to think of a positive number, triple the number, square the results and then add 7. If Jerry’s answer is 43, what was the original number? Use the concept of the inverse function and its domain and range in your solution.

Answer Key

What I Know

1. B
2. D
3. D
4. C
5. A
6. B
7. D
8. B
9. A
10. D
11. D
12. A
13. C
14. B
15. A

Assessment

1. A
2. B
3. D
4. C
5. C
6. C
7. D
8. B
9. A
10. D
11. D
12. A
13. C
14. B
15. A

What’s More

1. f(x) = 2x − 1

f⁻¹ =	(x + 1)/2
Domain	{x ∈ R}
Range	{y ∈ R}

2. f(x) = 5x − 2

f⁻¹ =	(x + 2)/5
Domain	{x ∈ R}
Range	{y ∈ R}

3. f(x) = (x + 2)/5

f⁻¹ =	5x − 2
Domain	{x ∈ R}
Range	{y ∈ R}

4. f(x) = x² + 2

f⁻¹ =	√(x − 2)
Domain	{x ≥ 2}
Range	{y ∈ R}

5. f(x) = √(1 + x)

f⁻¹ =	x² − 1
Domain	{x ∈ R}
Range	{y ≥ −1}

References

Alday, Eward M., Batisan, Ronaldo S., and Caraan, Aleli M. General Mathematics. Makati City: Diwa Learning Systems Inc.,2016.67-68.

Oronce, Orlando A., and Mendoza, Marilyn O. General Mathematics. Quezon City:Rex Bookstore, Inc.,2016.32-39.

Oronce, Orlando. General Mathematics. Quezon City:Rex Bookstore, Inc.,2016. 40-51

General Mathematics Learner’s Material. First Edition. 2016. pp. 77- 81

*DepED Material: General Mathematics Learner’s Material

For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resources (DepEd-BLR)

Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph *