This is a Self-Learning Module (SLM) for Senior High School General Mathematics, Quarter 1, Module 12: The Inverse of One-to-one Functions. It has one lesson. It shows learners how to find the inverse of a one-to-one function in two ways: by intuition, using the idea of “undo,” and by following a set of steps.
The module was developed by DepEd Region IV-A CALABARZON (First Edition, 2020) and written by Azalea A. Gallano.
On this page 7 sections
What learners will learn
After going through this module, learners are expected to:
- determine the inverse of a one-to-one function; and
- write a letter to a family member or peer about making amends for regretful events that cannot be undone.
Lessons in this module
- Lesson 1: The Inverse of a One-to-one Function. It defines an inverse function and explains why only a one-to-one function has an inverse that is also a function. It then works through linear, quadratic, and rational examples.
Key ideas: what an inverse function is
The module opens with a mind game. Think of a number, multiply it by 2, subtract 1, then add 4. Subtracting 1 and adding 4 is the same as adding 3, so the final answer is 2x + 3. To get back the original number, you “undo” the steps in reverse order, the same way the undo button on a phone or laptop removes your latest change first:
| Commands | Undo |
|---|---|
| Think of a number (x) | Start with the answer (y) |
| Multiply it by 2 (2x) | Subtract 3 (y − 3) |
| Add 3 (2x + 3) | Divide by 2: (y − 3)/2, which gives back x |
The module’s formal points:
- If a function has domain A and range B, its inverse is a function with domain B and range A.
- The inverse of f is written f⁻¹. By definition, f⁻¹(y) = x if and only if f(x) = y, for any y in range B.
- A function must be one-to-one for its inverse to also be a function. The inverse of a many-to-one function is one-to-many, which is not a function.
Key ideas: how to find the inverse
Method 1: By intuition (“undo”)
Undo each operation, starting with the last one. In Example 1, f(x) = 3x − 8:
- The last operation is subtraction, so add 8: x + 8.
- The operation before that is multiplication, so divide by 3: (x + 8)/3.
- Write the result as the inverse: f⁻¹(x) = (x + 8)/3.
Method 2: The general steps
The intuitive method is not easy in every case, so the module gives these steps:
- Express the function in the form y = f(x).
- Interchange the x and y variables.
- Solve for y in terms of x.
Worked examples from the module
- Example 2: g(x) = x² − 6x − 7. This is a quadratic function. Its graph is a parabola opening upward, so it fails the horizontal line test and is not one-to-one. Solving with the steps gives y = ±√(x + 16) + 3. Some x-values give two y-values, so g has no inverse function.
- Example 3: h(x) = (4x + 8)/(x − 3). Interchange x and y to get x = (4y + 8)/(y − 3). Multiply both sides by (y − 3), gather the y terms, and factor. The result is h⁻¹(x) = (3x + 8)/(x − 4).
In the solutions, APE and MPE stand for the addition and multiplication properties of equality.
How to use this module
The module follows the usual SLM parts:
- What I Know: a 15-item multiple-choice pretest on inverse operations, the symbol f⁻¹(x), and finding inverses.
- What’s In: a review where learners classify functions as linear, quadratic, or rational, and as one-to-one or many-to-one.
- What’s New: the “I Can See Your Mind” number game.
- What is It: the definition, the two methods, and Examples 1 to 3.
- What’s More: Activity 1 has five functions to invert by intuition. Activity 2 has ten functions to invert “if it exists.”
- What I Have Learned: four review questions, such as “Do all kinds of functions have inverse function?”
- What I Can Do: show that f(x) = |5x| has no inverse function, presented as a comic strip. It is scored on solution, strategy, and presentation.
- Assessment: a 15-item multiple-choice post-test.
- Additional Activities: write a letter of amends to a family member or peer and score it with the rubric. If the letter scores 15 to 20, give it. If it scores below 15, revise it first.
An answer key for the pretest, activities, and assessment is at the end.
For learners: Answer on a separate sheet of paper and do not write on the module. Use the answer key honestly, only to check your work.
For parents: Help your child follow the parts in order. Try the mind game together, as the module suggests.
For teachers: The Notes to the Teacher ask you to provide solutions for exercises learners cannot answer, especially in the What I Can Do part.
Frequently asked questions
What grade and quarter is this General Mathematics module for?
It is a Senior High School General Mathematics module for Quarter 1. It is Module 12, The Inverse of One-to-one Functions.
Which functions have an inverse function?
Only one-to-one functions have an inverse that is also a function. A many-to-one function, such as the quadratic g(x) = x² − 6x − 7, has an inverse that is one-to-many, which is not a function.
What are the steps to find the inverse of a one-to-one function?
Write the function as y = f(x), interchange x and y, then solve for y in terms of x. Simple functions can also be inverted by intuition, undoing each operation in reverse order.
Does the module have an answer key?
Yes. The answer key at the end covers the What I Know pretest, Activities 1 and 2, and the Assessment.
What is the letter-writing activity about?
In the Additional Activities, learners write a letter to a family member or peer expressing regret over an event and making amends. They score it with a rubric on content and on grammar and mechanics.
Full text of the module
The text below is a transcription of the PDF above, as printed.
General Mathematics Quarter 1 – Module 12: The Inverse of One-to-one Functions
Senior High School
[illustration: students studying under a tree and at a table, with a globe, a lamp, and a teacher]
General Mathematics Alternative Delivery Mode Quarter 1 – Module 12: The Inverse of One-to-one Function 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: Azalea A. Gallano
Editors: Elizabeth B. Dizon, Anicia J. Villaruel, and Roy O. Natividad
Reviewers: Fritz A. Caturay, Necitas F. Constante, Dexter M. Valle, and Jerome A. Chavez, January B. Regio, John Lester F. Guerero 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
Senior High School
General Mathematics Quarter 1 – Module 12: The Inverse of One-to-one Functions
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
[illustration: hand pointing with a finger]
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:
- determine the inverse of a one-to-one function.
- write a letter to a family member or peer about making amends on regretful events which cannot be undone.
What I Know
[illustration: hand holding a pencil]
Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.
Which is related to the word “inverse”?
a. cancel
b. change
c. delete
d. interchangeWhat is the inverse of addition?
a. division
b. multiplication
c. subtraction
d. compositionWhat is the inverse of division?
a. addition
b. multiplication
c. subtraction
d. compositionWhat is the inverse of multiplication?
a. addition
b. division
c. subtraction
d. compositionWhat is the inverse of subtraction?
a. addition
b. division
c. multiplication
d. compositionWhich of the following functions has/have inverse function?
a. one-to-one
b. many-to-one
c. both
d. noneWhat is the mathematical symbol for inverse of f(x)?
a. 1/f(x)
b. f(x)⁻¹
c. ′f(x)
d. f⁻¹(x)What is the inverse of f(x) = 3x – 4?
a. f(x) = 3x + 4
b. f(x) = 4x – 3
c. f(x) = (x + 3)/4
d. f(x) = (x + 4)/3What is the inverse of f(a) = a³ + 5?
a. f(a) = 3a – 5
b. f(a) = ∛(a – 5)
c. f(a) = ∛a – 5
d. f(a) = a/3 + 5Which is a property of an inverse function?
a. The inverse of f⁻¹(x) is −f⁻¹(x).
b. f⁻¹(f(x)) = x for all positive x in the domain of f.
c. f⁻¹(f⁻¹(x)) = x for all x in the domain of f⁻¹.
d. f(f⁻¹(x)) = x for all x in the domain f⁻¹.What is the inverse of f(x) = -2x + 7?
a. f(x) = −(x – 7)/2
b. f(x) = −x/2 + 7
c. f(x) = 7x – 2
d. f(x) = 2x – 7What is the inverse of f(b) = b³ + 4?
a. f(b) = 3b – 4
b. f(b) = b/3 + 4
c. f(b) = ∛b + 4
d. f(b) = ∛(b – 4)What is the inverse of g(x) = 2x – 8?
a. g′(x) = (x + 8)/2
b. g′(x) = −2x + 8
c. g⁻¹(x) = (x + 8)/2
d. g⁻¹(x) = −2x + 8What is the inverse function of f(c) = (2c + 1)/(4c – 5)?
a. f⁻¹(c) = (4c – 1)/(2c + 5)
b. f⁻¹(c) = (5c + 1)/(4c – 2)
c. f′(c) = (2c – 1)/(4c + 5)
d. f′(c) = (4c – 5)/(2c + 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.
Lesson 1 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
[illustration: hand holding a puzzle piece]
So far, you have known different faces of functions in the previous lessons. Likewise, you have categorized them already into groups of one-to-one and many-to-one functions. Let’s have a quick review!
Directions: 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 |
|---|---|---|
| 1. f(x) = 2x + 5 | ||
| 2. g(x) = 2x² – 4x + 1 | ||
| 3. h(x) = (x – 1)² – 1/2 | ||
| 4. f(x) = (2x – 1)/(x + 5) | ||
| 5. g(x) = x |
Do you ever wonder if inverses of these functions are functions as well? Well, both the original equation and its inverse are both functions. In this lesson, you will delve into these functions with function inverses.
[illustration: hand holding a stack of papers]
Notes to the Teacher
Solutions should be provided for exercises which will not be successfully answered by the learners especially for “What I Can Do” Part.
What’s New
I Can See Your Mind
Let’s have a mind game. Ready?
Think of a number and 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. $\frac{y-3}{2}$ |
| Step 1. Think of a number. (x) | Answer will be the number you are thinking. (x) |
By that way, I have seen your mind and have created 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 $f^{-1}(y) = x$ 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.
How to Find the Inverse of One-to-one Function
Intuitively, the inverse of a function may be known by the principle of “undo”. That is, by considering the inverses of the operations performed, the inverse of a function may be computed easily.
Example 1
Given $f(x) = 3x – 8$, the inverse of a function may be solved intuitively.
Solution:
| Steps | In symbols |
|---|---|
| Step 1. The last operation performed is subtraction, the inverse operation of which is addition. To x, add 8. | x + 8 |
| Step 2. The second to the last operation performed is multiplication, the inverse operation of which is division. Divide x + 8 by 3. | $\frac{x+8}{3}$ |
| Step 3. Equate it to $f^{-1}(x)$ to denote that it is the inverse function of $f(x) = 3x – 8$. | $f^{-1}(x) = \frac{x+8}{3}$ |
However, it is not that easy in some cases. In later examples, you will understand what I mean by saying that there is a more general method that may be followed.
To find the inverse of a one-to-one function, consider the following steps:
a. Express the function in the form $y = f(x)$; b. Interchange the x and y variables in the equation; c. Solve for y in terms of x.
Example 2
If it exists, solve for the inverse of $g(x) = x^2 – 6x – 7$.
Solution:
Recognize that $g(x)$ is a quadratic function whose graph is a parabola opening upward. It fails the horizontal line test because it has x-values which correspond to the same y-value. And since it is not a one-to-one function, then its inverse is not a function. Simply put, it has no inverse function.
Alternate Solution:
$y = x^2 – 6x – 7$ (change g(x) to y)
$x = y^2 – 6y – 7$ (interchange x and y)
$y^2 – 6y = x + 7$ (solve for y, APE)
$y^2 – 6y + 9 = x + 7 + 9$ (solve for y, by completing the square, by APE)
$(y – 3)^2 = x + 16$ (solve for y, by factoring)
$y – 3 = \pm\sqrt{x + 16}$ (solve for y, by getting the square root of both sides)
$y = \pm\sqrt{x + 16} + 3$ (solve for y, by APE)
Notice that for some values of x, there are two values of y. For instance, if x=1, $y = \sqrt{17} + 3$ and $y = \sqrt{17} + 3$. [transcriber’s note: the two values are printed identically in the original; the second may have been meant to carry a minus sign, as in the ± above, but it is transcribed as printed] Therefore, the inverse function of $g(x)$ does not exist.
Example 3
Find the inverse of the rational function $h(x) = \frac{4x+8}{x-3}$.
Solution:
$y = \frac{4x+8}{x-3}$ (change g(x) to y)
$x = \frac{4y+8}{y-3}$ (interchange x and y)
$xy – 3x = 4y + 8$ (solve for y, MPE)
$xy – 4y = 3x + 8$ (solve for y, by APE)
$y(x – 4) = 3x + 8$ (solve for y, by factoring)
$y = \frac{3x+8}{x-4}$ (solve for y, by MPE)
$h^{-1}(x) = \frac{3x+8}{x-4}$ (the inverse function)
What’s More
Activity 1
Directions: By intuition, give the inverse function of each of the following. Write your answers on a separate sheet of paper.
- $f(x) = x + 2$
- $g(x) = 12x – 1$
- $h(x) = -\frac{x}{4}$
- $f(x) = x$
- $g(x) = \frac{3x+5}{8}$
Activity 2
Directions: Solve for the inverse function of each of the following if it exists.
- $f(x) = 25x – 18$
- $g(x) = \frac{12x-1}{7}$
- $h(x) = -\frac{9x}{4} – \frac{1}{3}$
- $f(x) = x^9$
- $f(a) = a^3 + 8$
- $g(a) = a^2 + 8a – 7$
- $f(b) = (b + 6)(b – 2)$
- $h(x) = \frac{2x+17}{3x+1}$
- $h(c) = \sqrt{2c + 2}$
- $f(x) = \frac{x+10}{9x-1}$
What I Have Learned
Answer the following questions.
- What is an inverse function?
- What is the symbol of an inverse function?
- Do all kinds of functions have inverse function?
- How do you solve for the inverse of a one-to-one function?
What I Can Do
Show that $f(x) = |5x|$ has no inverse function. Present your work through a comic strip. Use a separate sheet of paper for your work.
The following rubric will be used in rating your work:
| Criteria | 3 | 2 | 1 |
|---|---|---|---|
| Solution (score x 4) | with correct solution | with minor computational errors | with major computational errors |
| Strategy (score x 3) | with insightful strategy that demonstrates conceptual understanding of mathematical rules and properties | with appropriate strategy that demonstrates procedural understanding, but not conceptual understanding | with absence of logic in strategy that will not lead to the correct answer |
| Presentation (score x 3) | images and captions show an exceptional degree of creativity; with clear identification of characters | images and captions show some degree of creativity; with identification of some characters | images and captions show a little degree of creativity; characters cannot be clearly identified |
Assessment
Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.
Which relates to “inverse”?
a. redo
b. opposite
c. delete
d. interchangeWhat is the inverse of $f(x) = x + 36$?
a. $f^{-1}(x) = \frac{x}{36}$
b. $f^{-1}(x) = 36x$
c. $f^{-1}(x) = x – 36$
d. $f^{-1}(x) = -x – 36$What is the inverse of $f(x) = \frac{x}{25}$?
a. $f^{-1}(x) = x + 25$
b. $f^{-1}(x) = 25x$
c. $f^{-1}(x) = x – 25$
d. $f^{-1}(x) = -25x$What is the inverse of $f(x) = -3x$?
a. $f^{-1}(x) = x + 3$
b. $f^{-1}(x) = -\frac{x}{3}$
c. $f^{-1}(x) = x – 3$
d. $f^{-1}(x) = \frac{x}{3}$What is the inverse of $f(x) = x – 10$?
a. $f^{-1}(x) = 10 + x$
b. $f^{-1}(x) = -\frac{x}{10}$
c. $f^{-1}(x) = -10x$
d. $f^{-1}(x) = -(x – 10)$Which characterizes an inverse function?
a. Given that it is $f^{-1}(x)$, its domain and range are the same as the domain and range of $f(x)$.
b. It is denoted by $y = f^{-1}(x)$.
c. Its inverse is one-to-one or many-to-one.
d. It is one-to-one or many-to-one.What is the mathematical symbol for inverse of $f(x)$?
a. $\frac{1}{f(x)}$
b. $f(x)^{-1}$
c. $’f(x)$
d. $f^{-1}(x)$What is the inverse of $f(x) = -2x – 8$?
a. $f(x) = 8x + 2$
b. $f(x) = 2x + 8$
c. $f(x) = \frac{x+2}{8}$
d. $f(x) = \frac{x+8}{-2}$What is the inverse of $f(x) = b^5 + 2$?
a. $f(x) = 2b – 5$
b. $f(x) = 5b + 2$
c. $f(x) = \sqrt[5]{b-2}$
d. $f(x) = \sqrt[5]{b} – 2$Which is a property of an inverse function?
a. The inverse of $f^{-1}(x)$ is $f(x)$.
b. $f^{-1}(f(x)) = x$ for all negative x in the domain of $f$.
c. $f^{-1}(f^{-1}(x)) = x$ for all x in the domain of $f^{-1}$.
d. $f(f^{-1}(x)) = x$ for all positive x in the domain of $f^{-1}$.What is the inverse of $f(x) = -6x – 5$?
a. $f(x) = \frac{x+5}{-6}$
b. $f(x) = -\frac{x}{6} + 5$
c. $f(x) = 6x + 5$
d. $f(x) = 5x + 6$What is the inverse of $f(c) = (c + 1)^3 – 1$?
a. $f(c) = \sqrt[3]{c}$
b. $f(c) = \sqrt[3]{c + 1} – 1$
c. $f(c) = \sqrt[3]{c – 1} + 1$
d. $f(c) = 1 – \sqrt[3]{c + 1}$What is the inverse of $g(x) = 9x + 20$?
a. $g'(x) = \frac{x+20}{-9}$
b. $g'(x) = -20x + 9$
c. $g^{-1}(x) = \frac{x-20}{9}$
d. $g^{-1}(x) = -9x – 20$What is the inverse function of $f(d) = \frac{d-12}{2d+1}$?
a. $f'(d) = \frac{d+12}{-2d-1}$
b. $f'(d) = \frac{-d+12}{-2d-1}$
c. $f'(d) = \frac{d-12}{2d-1}$
d. $f'(d) = \frac{-d-12}{2d-1}$Which is involved in the process of finding the inverse of a function?
a. Write the function in the form $x = f(x)$.
b. Cancel the x variable.
c. Write in the function in the simplest form.
d. Solve for y in terms of x.
Additional Activities
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 (score x 3) | 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 (score x 2) | 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 and it ranged from 15 to 20, proceed giving your letter wholeheartedly. If the score you give is below 15, consider revising it before giving it to your loved one. This is a rare moment, make it count.
[transcriber’s note: this page is printed upside down; the columns are transcribed in logical order, and the “Assessment” column is placed last. Some mathematical expressions are uncertain and are marked [?].]
Answer Key
What I Know
- D
- C
- B
- B
- A
- A
- D
- D
- B
- D
- A
- D
- C
- B
- C
What’s More
Activity 12.1
- f^-1(x) = x − 2
- g^-1(x) = 12/(x + 1)
- h^-1(x) = −4x [?]
- f^-1(x) = x
- g^-1(x) = (8x − 5)/3
Activity 12.2
- f^-1(x) = (x + 18)/25
- g^-1(x) = (7x + 1)/12
- h^-1(x) = 9/(x/3 − 4) or −(4x/9) + 27/4 [?]
- f^-1(x) = ⁹√x [?]
- f^-1(a) = ∛(a − 8) [?]
- It has no inverse. Initially, the given is one-to-one 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.
- It has no inverse. Initially, the given is one-to-one 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.
- h^-1(x) = (−x + 17)/(3x − 2) or (−x − 17)/(3x − 2) [?]
- h^-1(c) = (c² − 2)/2
- f^-1(x) = (x + 10)/(9x − 1) [?]
Assessment
- D
- C
- B
- B
- A
- B
- D
- D
- C
- A
- A
- B
- C
- D
- D
Reference
Verzosa, Debbie Marie, et.al. 2016. General Mathematics: Learner’s Material, First Edition. Philippines: Lexicon Press Inc.
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