1

SENIOR HIGH SCHOOL

General Mathematics

Quarter 1 – Module 3 Week 4:One-to-One and

Inverse Functions

Department of Education Republic of the Philippines

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General Mathematics – Grade 11

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Quarter 1– Module 3: One-to-One and Inverse Functions

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SENIOR HIGH SCHOOL

3

SENIOR HIGH S SENIOR HIGH SCHOOL

General Mathematics

Quarter 1 – Module 3 Week 4:One-to-One and

Inverse Functions

This instructional material was collaboratively developed and

reviewed by educators from public and private schools, colleges, and

or/universities. We This instructional material was collaboratively developed and

reviewed encourage teachers and other education stakeholders to email their

feedback, comments, and recommendations to the Department of Education at

action@deped.gov.ph.We value your feedback and recommendations..

1

What I know

Multiple Choice. Write the letter of

your answer on a separate sheet of paper.

1. It is a relation where each element in the domain is related to only one value in the range by some rule.

A. One-to-one function C. Function

B . Relation D. Many-to-one function

2. What do call the rule that relates values from a set of values (called the domain) to a second set of values (called the range)?

A. One-to-one function C. Function

B. Relation D. One-to-many relatiom

3. What is the inverse of f(x)=1

𝑥 ?

A. 1

𝑦 C. 1

B. B. 1

𝑥 D.

𝑥

1

4. What do you call a line test used to determine if the given function is a one-to-one function?

A. Line bar test C. Curve line test

B. Horizontal line test D. Vertical line test

5. The graph of the inverse of a function can be obtained by reflecting the graph of the function along ___.

A. y=x C. X-axis

B. y-axis D. origin

6. What type of functions do not contain the same y-values to be paired in different two x-values.

A. Rational Function C. one-to-one function

B. Linear function D. One-to-many

7. Which of the following is NOT a one-to-one Function? A. {(1,2),(3,4),(5,6),(7,8)} C. {(1/2,1),(1/3,2),(2/3,3),(1/4,4)

B. {(-1,1),(-2,2),(-3,3),(-4,4)} D. {(1,1),(2,2),(3,3),(4,4)}

2

8. All x values or inputs are called what? A. Range C. Domain

B. Relation D. Function

9. All y values or outputs are called what? A. Range C. Domain

B. . Relation D. Function

10. Which of the following ordered pair is NOT a function? A. {(1,2),(2,1),(3,4),(4,3)} C. {(1,2),(1,1),(5,4),(4,3)}

B. {(1,1),(2,2),(3,3),(4,4)} D. {(-1,1),(1,2),(-3,3),(-4,4)}

For numbers 11 to 15. Given the function f(x)= 4𝑥+3

2𝑥−1.

11. Find the domain of a function.

A. {x є R | x ≠ - 1

2} B. {x є R | x ≠

1

2}

C.{x є R | x ≠ - 2} D. {x є R | x ≠ 2}

12. Find the range of a function.

A. {y є R | y ≠ - 1

2} B. {y є R | y ≠

1

2}

C. {y є R | y ≠ - 2} D. {y є R | y ≠ 2}

13. Find the inverse of a function.

A. f-1 = 𝑥−3

2𝑥−4 C. f-1 =

2𝑥−4

𝑥−3

B. f-1 = 𝑥+3

2𝑥−4 D. f-1 =

𝑥+3

2𝑥+4

14. Find the domain of the inverse function.

A. {x є R | x ≠ - 1

2} B. {x є R | x ≠

1

2}

C.{x є R | x ≠ - 2} D. {x є R | x ≠ 2}

15. Find the range of the inverse function.

A. {y є R | y ≠ - 1

2} C. {y є R | y ≠ - 2}

B. {y є R | y ≠ 1

2} D. {y є R | y ≠ 2}

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What I need to know

At the end of the lesson, the learner will be able to:

Represent real life situation using one-to-one

Illustrate one-to-one functions

Identify one-to-one functions

Illustrate horizontal line test

What’s in…

Activity 1.

Identify whether each relation represents a function or not?

1. {(1,2), (3,4), (5,6), (7,8), (9,10)}

2. {(3,2), (2,4), (1,6), (5,3), (6,3)}

3. {(2,2), (4,4), (3,2), (3,7), (1,4)}

For 4 & 5 evaluate the following functions at x = 3.

4. f = 5x + 3

5. g = 2𝑥−1

𝑥+2

Activity 2.

Identify whether the given situation represents one-to-one function. Justify your

answer. Write your answer on a separate sheet of paper.

1. The relation pairing the LRN to students 2. The relation pairing a real number to its square.

Lesson

1

One-to-One Function

What’s new…

4

3. {(3,1), (4,2), (3,2), (1,2), (5,4)} 4. {(2,2), (4,4), (3,2), (5,7), (1,4)} 5. {(1,2), (3,4), (5,6), (7,8), (9,10)}

Questions:

1. How did you find the activity?

2. How are you able to connect this activity in your daily life?

In the preceding activity, item numbers 1 and 5 represents a one-to-one function

since each student has a unique LRN and in item number 5, there is no y – value which is

paired with the same x – values.

Item numbers 2 to 4 does not represent a one-to-one function. In item number 3, the

value of domain is repeated (3,1) and (3,2). Also in number 4, it has a y-value that are paired

up with two different x-values; ( 2,2 ) and ( 3,2 ).

Another way to determine whether a graph represents a one-to-one function is by

using the Horizontal Line Test and Vertical Line Test.

Horizontal Line Test is a test used a determine a one-to-one function. All one-

to-one functions satisfy both the vertical and horizontal line tests.

A graph showing the plot of y= x2 + 2 fails the horizontal line test If any vertical line

and horizontal line intersects the graph at most 1 point, then the graph is a graph of a one-

to-one function. The graph of y = x3 passes the horizontal line test.

y = x3 y = x2 + 2

One-to-one function Not a one-to-one function

One-to-One Function

The function is one-to-one if for any x1 , x2 in the domain of f, then f(x1) ≠ f(x2) .

That is, the same y – value is never paired with two different x – values.

What is it

5

What’s more

Activity 3.

Which of the following relation is a one-to-one function? Write your

answers on a separate sheet of paper.

(a) {(0,0),(1.1),(2,8),(3,27),(4,64)}

(b) {(-2,4),(-1,1),(0,0),(1,1),(2,4)}

(c) {(0,4),(1,5),(2,6),(3,7),..(n,n+4),…}

(d) Height to student

(e) Person to telephone number

(f) Height to age

(g) Birthdate of a family member

What have I have learned..

Activity 4: Question and Answer

Directions: Answer the questions briefly. Write your answers on a separate

sheet of paper.

1. What is a One-to-One Function?

2. How to identify a one-to-one function?

3. What is horizontal line test?

4. What is vertical line test?

5. How will we know that a given graph is a graph of a one-to-onee function?

Activity 5.

1. Give real life situations that represents one-to-one function. 2. How can we apply the concept of one-to-one function in daily life?

What I can do

6

Activity 6.

Answer as indicated. Write your answers on a separate sheet of paper.

1. Write your own example of a one-to-one function and represent in terms of

ordered pairs, graph, and table of values.

2. Why do we need to study about one-to-one function?

Additional Activities

7

What I need to know

At the end of the lesson, the learner will be able to:

Illustrate the inverse of a one-to-one functions

Solve the inverse of a one-to-one functions

State the properties of inverse functions

What’s in

Activity 1.

Identify whether the given situation represents one-to-one function or Not.

1. Sim cards to cellphone numbers

2. Pairing of airports to airport code

3. The function of f (x)= x2

4. {(1,2),(1/2,3),(1/3,4),(1/4,5),(1/5,6)}

5. {(2,1),(1,3),(4,8),(5,7),(6,9)}

6.

7.

Lesson

2

Inverse of a One-to-

One Function

x

y

x

y

8

Activity 2.

Match the following one-to-one function to its inverse. Write your answers on a

separate sheet of paper.

A B

1. f(x) = 3x+1 a. y=

2. g(x) = x3-2 b. y=

3. f(x) = c. y=3√(𝑥 + 2)

4. f(x) = x2+4x-2 d. y= 𝑥−1

3

Questions:

1. How did you get your answer?

2. Is it easy to evaluate inverse of one-to-one function?

3. What makes you difficult in answering functions?

What is it

Inverse of a One-to-One Function

Let f be a one-to-one function with domain A and a range B. The inverse of f denoted by f-1 is a function with a domain B and a range A defined by f-1(y) = x, if and only if f(x) = y , for any y in B.

Only one-to-one function has an inverse . If a function f is not one-to-one, properly

defining an inverse function f-1 will be problematic. For example, suppose that f(1) = 5 and

f(3) = 5. If f-1 exists, then f-1 (5) has to be both 1 and 3, and this prevents f-1 from being a valid

function. This is the reason why the inverse is only defined for one-to-one functions.

In the previous activity, we can find the inverse of a function by following the given steps.

(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

What’s new…

9

In Item number 1, to find the inverse of f(x) = 3x+1, we have these step by step solutions:

(a) The equation of the function is y=3x+1 (b) Interchange the x and y variables: x=3y+1 (c) Solve for y in terms of x: x=3y+1

x-1=3y

𝑥−1

3= 𝑦

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

3

In item number 2, the given is g(x) = x3-2, solving for its inverse, we have

y= x3-2. x= y3 – 2 x= y3-2

x+2=y3

𝑦 = √𝑥 + 23

Therefore, the inverse of g(x)= x3-2 is g-

Using the same steps, the resulting inverse of y = 2𝑥+1

3𝑥−4 is is f-1 (x) =

4𝑥+1

3𝑥−2

In item number 4, the given function y=x2+4x-2 is not one-to one since it is a quadratic function and the graph is a parabola that opens upward. This implies that these function fails the horizontal line test. Thus, it has no inverse.

For the second and third properties above, it can be imagined that evaluating a

function and its inverse in succession is like reversing the effect of the function. For

example, the inverse of a function that multiplies 3 to a number and adds 1 is a function that

subtracts 1 and then divides the result by 3.

Examples: 1. 𝑓 = 2𝑥 + 4 ; 𝑓−1 =𝑥−4

2

2. 𝑓 =3𝑥+4

2 ; 𝑓−1 =

2𝑥−4

3

3. 𝑓 =3𝑥+4

2𝑥−2 ; 𝑓−1 =

2𝑥+4

2𝑥−3

Property of an inverse of a one-to-one function

Given a one-to-one function f(x) and its inverse f-1(x), then the following are true:

(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 domain of f.

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What’s more

Activity 3. Apply the steps in solving the inverse of a one-to-one function? Write your answers in a separate sheet of paper.

1. f(x)=- 1

3𝑥 + 1

2. f(x)= 2x-1

3. g(x)= 3𝑥−1

2𝑥+3

What have I have learned

Activity 4: Question and Answer

Directions: Answer the questions briefly. Write your answers in a separate sheet of paper.

1. What is an inverse function?

2. How to identify an inverse of a one-to-one function?

3. What are the steps in solving the inverse of a one-to-one function? 4. State the properties of an inverse function. Write your answers on a separate

sheet of paper.

Activity 5.

Give the inverse of the following functions if it exists? Write your answers in a

separate sheet of paper.

1. f(x)=|5x|

2. f(x)=x2-4

3. g(x)=

What I can Do

11

Answer the following problem. State the properties of inverse function in

every steps. Write your answers in a separate sheet of paper.

1. Find the inverse of f(x)= 4x+2

2. Find the inverse of h(x)= x2-4

3. Find the inverse of g(x)=𝑥+5

𝑥−5

4. Find the inverse of f(x)=−2𝑥+7

𝑥

Additional Activities

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At the end of the lesson, the learner will be able to:

Represent an inverse function through its table of

values and graph

Find the domain and range of an inverse function

Solve problems involving inverse functions

Activity 1.

A. Make a table of values representing the following functions when x

ranges from -2 to 3. Draw the graph .

1. y = x + 3

2. f(x)=2x+1

3. ƒ(x)= 1

𝑥

B. Give the inverse of the functions in A.

He follow

Activity 1.

Represent the following inverse functions in table of values with the

specified domain.

Lesson

3

Graphs of an Inverse

Function

What’s new…

What I need to know

What’s in

13

1. f-1 (x) = 𝑥−1

2 ; {x|-2 ≤ x ≤ 1.5}

2. f-1 (x) = x – 4 ; {x|-2 ≤ x ≤ 2}

Activity 2.

Using the resulting table of values represented in the preceding activity,

(Activity 1), determine the domain and the range.

1. f-1 (x) = 𝑥−1

2 ; domain:_______________

Range:________________

2. f-1 (x) = x – 4 ; domain:_______________

Range:________________

Activity 3.

Given are the one-to-one functions and its graph. Beside, is the inverse of each

function.Sketch the graph of the corresponding inverse function and determine the domain

and range.

Function Inverse Function

1. f(x)=2x+1 ; f-1 (x) = 𝑥−1

2

Domain of f-1 (x) ;__________________________

Range of f-1 (x) ; ___________________________

2. g(x) = √𝑥 + 13

; g-1 (x) = x3 – 1

14

Domain of g-1 (x) ;__________________________

Range of g-1 (x) ; ___________________________

Graphs of an inverse function

First, we need to ascertain that the given graph corresponds to a one-to-one

function by applying the horizontal line test. If it passes the test, the corresponding function

is one-to-one.

In the preceding activity, the graph of the inverse of f(x)=2x+1 is shown below.

Domain of f-1 (x) ; D ={ x є R |-3≤ x ≤ 4}

Range of f-1 (x) ; R = { y є R |-2≤ y ≤ 1.5}

In the same manner, the graph of the inverse of g-1 (x) = x3 – 1 is shown below.

What is it

f-1 (x) = 𝑥−1

2

f(x)=2x+1

x3 – 1

15

Domain of g-1 (x) ; D = {x/-2 ≥ x ≥ 2}

Range of g -1 (x) ; R = all real numbers

Table summary,

ƒ(x) ƒ-1(x)

Domain {x є R |-2 ≤ x ≤ 1.5} { x є R |-3≤ x ≤ 4}

Range { y є R |-3≤ y ≤ 4} { y є R |-2≤ y ≤ 1.5}

g(x) g -1(x)

Domain All real numbers {x/-2 ≥ x ≥ 2}

Range {y/-2 ≥ y ≥ 2} All real numbers

Solving Problems Involving Inverse Function

We can apply the concepts of inverse functions in solving word problems involving reversible processes.

Example:

You asked a friend to think of a nonnegative number, add two to the number, square

the number, multiply the result by 3 and divide the result by 2. If the result is 54, what is the

√𝑥 + 13

Given the graph of a one-to-one function, the graph of its inverse can be

obtained by reflecting the graph about the line y = x.

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

inspection of the graph.

16

original number? Construct an inverse function that will provide the original number if the

result is given. https://www.scribd.com/document/413943141/Gen-Math-Pre-Test

Questions:

1. What are your ideas in solving this problem?

2. In what part of the problem makes you difficult in arriving the answer?

3. Is the problem easy? Why?

We first construct the function that will compute the final number based on the

original number. Following the instructions, we come up with this function:

3(𝑥+2)2

2

ƒ(x) = (x+2)2 ▪ 3 ÷ 2

The graph is shown below, on the left. This is not a one-to-one function because the

graph does not satisfy the horizontal line test. However, the instruction indicated that the

original number must be nonnegative. The domain of the function must thus be restricted to

x ≥ 0, and its graph is shown on the right, below.

The function with restricted domain x ≥ 0 is then a one-to-one function, and

we can find its inverse.

Interchange the x and y variables: x =

Solve for y in terms of x:

x = 3(𝑦+2)2

2

https://www.scribd.com/document/413943141/Gen-Math-Pre-Test

17

√2𝑥

3 = y+2

(Since y ≥ 2, we do not need to consider the negative value of √2𝑥

3 )

2 = y → y = 2 → ƒ-

Finally, we evaluate the inverse function at x = 54 to determine the original number

ƒ- 2 = 6 – 2 = 4

the original number is 4.

What’s more..

Activity 4.

1. Find the inverse of ƒ(x)= 1

𝑥 using its given graph.

2. Consider the rational function ƒ(x)= whose graph is shown below:

18

(a) Find its domain and range.

(b) Find the graph of its inverse.

(c) Find the domain and range of its inverse.

What have I have learned..

Activity 5:

Answer the questions briefly. Write your answers in a separate sheet of

paper.

1. What is the graph of inverse function?

2. How to identify the domain and range of the graph of inverse

function?

Activity 6.

Solve the problem .

Given ƒ(x)=−4𝑥−1

3𝑥−2

(a) Solve its inverse

(b) Determine the asymptotes

What I can do

19

(b) Sketch the graph of its inverse

(c) Find the domain and range of its inverse.

Read the problem carefully and write your answer on the separate sheet of

paper.

1. A function is said to be a one-to-one function if __________.

A. The function ƒ is one-to-one if for any x1, x2 in the domain of f, then f(x1) = f(x2)

B. the y-value is paired with two different x-values

C. The function ƒ is one-to-one if for any x1, x2 in the domain of f, then f(x1) ≠ f(x2)

D. Two x-values are paired with 2 y-values

2. Which of the following belong to one-to-one function?

A. People to their birthdays

B. People to their place of residence

C. People to their GSIS number

D. People to their first name

3. Determine which of the graph describe a one-to-one function?

A. C.

B. D.

4. What is the collection of input values in a function called?

A. Ordered pair C. Domain

B. Range D. Y-values

5. Which of the following map describe a one-to-one function?

A. C.

Assessment…

20

B. D.

6. What is the inverse of f(x)= 2

3x + 1

A. 3𝑥−3

2 C.

2

3𝑥−3

B. D. 3𝑥−2

3

For numbers 7 to 12. Given the function of 𝑔(𝑥) =7𝑥+2

3𝑥−4

7. Find the domain of a function. A. {x є R | x ≠ -3/4 } C.{x є R | x ≠ 7/3} B. {x є R | x ≠ 4/3 } D. {x є R | x ≠ 3/7}

8. Find the range of a function.

A. {y є R | y ≠ -3/4 } C.{y є R | y ≠ 7/3} B. {y є R | y ≠ 4/3 } D. {y є R | y ≠ 3/7}

9. Find the domain of an inverse function.

A. {x є R | x ≠ -3/4 } C.{x є R | x ≠ 7/3} B. {x є R | x ≠ 4/3 } D. {x є R | x ≠ 3/7}

10. Find the range of an inverse function.

A. {y є R | y ≠ -3/4 } C.{y є R | y ≠ 7/3} B. {y є R | y ≠ 4/3 } D. {y є R | y ≠ 3/7}

11. Find the vertical asymptotes of inverse function.

A. ¾ C. 7/3 B. 4/3 D. 3/7

12. Find the horizontal asymptotes of inverse function.

A. ¾ C. 7/3

B. 4/3 D. 3/7

13. Find the inverse of h(x)= 3x + 6.

A. 1

3𝑥 + 2 C.

3

𝑥− 2

B. 1

3𝑥 − 2 D.

𝑥

3− 2

14. Which of the graph illustrate a one-to-one .

21

A. C.

B. D.

15. Which of the graph illustrate a one-to-one function using horizontal line test?

A. C.

B. D.

Activity 7.

Answer the problem. Write your answer in a separate sheet of paper.

Find the domain and range of the inverse of ƒ(x) = x2 – 6x + 5 with domain

restriction {x є R | 0 < x < 3}.

Additional Activities

22

Key Answers…

Lesson 1

23

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

Lesson 1.

What’s In

1.Function

2.Function

3.Not

4.18

5.1

What’s New

1.One-to-one

2.Not

3.Not

4.Not

5.One-to-one

What’s More

a, c, and e

What I have learned

1.The function is one-to-

one if for any x1 , x2 in

the domain of f, then

f(x1) ≠ f(x2) .

2.the same y – value is

never paired with two

different x – values.

3.a test used a

determine a one-to-one

function

4.A test used to

determine a graph of a

function

5.Each horizontal line

intersects the graph at

most one point

What Can I Do

- Answer varies

Additional Activities

- Answer varies

Lesson 2

What’s In

1.One-to-one

2.One-to-one

3.Not

4.One-to-one

5.One-to-one

6.Not

7.One-to-one

What’s New

1. D

2. C

3. B

4. a

What’s More

1.F-1

(x)=3(x-1)

2.F-1

(x)=(x+1)/2

3.F-1

(x)=(-3x-1)/(2x-3)

What I have Learned

1.The inverse of f

denoted by f-1

is a

function with a domain

B and a range A

defined by f-1

(y) = x, if

and only if f(x) = y , for

any y in B.

2.By following 3 steps

3.-Write the function in

the form y = f(x);

-Interchange the x and

y variables;

-Solve for y in terms of

x

4.The inverse of f-1(x) is

f(x).

-f(f-1(x))=x for all x in

the domain of f-1.

-f -1(f(x))=x for all x in

the domain of f.

What I Can Do

1.Invese does not exist

2.Inverse does not exist

3.G-1

(x)=(-4x-1)/(3x-2)

Additional Activities

1.F-1

(x)=(x-2)/4

2.h-1

(x)=√𝑥+4

3.g-1

(x)=(5x+5)/(x-1)

4.f-1

(x)=7/(x+2)

Lesson 3

What’s In

1.

X -2 -1 0 1. 2. 3.

y 1 2 3 4. 5. 6.

2.

X -2 -1 0 1. 2. 3.

y -3 -1 1 3 5 7

3.

X -2 -1 0 1. 2. 3.

y -1/2 -1 - 1 1/2 1/3

1.

3

-3

2.

3

-1/2

3.

24

References:

Teaching Guide for Senior High School-General Mathematics

By Commission on Higher Education (2016)

Pages: 60-75

https://quizizz.com/admin/quiz/57ee5e103803b30115d44287/relations-and-

functions

What’s new

1.

X -2 -1 0 0.5 1 1.5

y -3/2 -1 -1/2 -1/4 -0 1/4

2.

X -2 -1 0 1 2

y -6 -5 -4 -3 -2

What’s More

1.

2.Domain of ƒ(x)= { x є R |x ≠ 2}

Range of ƒ(x)= { y є R |y ≠ -5}

{ x є R | x ≠ -5}

{ y є R |y ≠ 2}

What i have learned

1.Reflection of the graph of a function along y=x

2.The domain of the function is the range of its

inverse and the range of the function is the domain

of its invese.

What I can do.

1.F-1

(x)= (2x-1)/(3x+4)

2.Vertical asymptote; x=-4/3

Horizontal asymptote; y= 2/3

3.

4. Domain= {x/ x≠ -4/3}

Range=(-∞,2/3)u(2/3,+∞)

Additional Activities

Domain: {x/-4≤x