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General Mathematics

General Mathematics

Quarter 1 – Module 14:

Domain and Range of Inverse

Functions

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 c. Real Numbers

b. Domain 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 c. decimal number

b. negative number d. irrational number

3. What is the domain and range of the function

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 (𝑥) = 3 𝑥 + 6?

a.. 𝑓

− 1

𝑥+ 6

3

c. 𝑓

− 1

𝑥− 6

3

b. 𝑓

− 1

3

𝑥− 6

d. 𝑓

− 1

3

𝑥+ 6

5. Which of the following pairs of functions is NOT the inverse of each other?

a. 𝑓(𝑥) = 2 𝑥 + 5 𝑎𝑛𝑑 𝑔(𝑥) = 2 𝑥 − 5

b.𝑓

𝑥

3

c. 𝑓

d.𝑓

For numbers 6-10, consider the function (𝑥) =

3

𝑥+ 1

6. What is the domain of the function?

a.

c.

b.

d.

7. What is the Range of the function?

a. {𝑦 ≠ 0 } c. {𝑦 ≠ 3 }

b. {𝑦 ≠ 1 } d. {𝑦 ≠ − 1 }

8. What is the inverse of the function?

a. 𝑓

− 1

3 +𝑥

𝑥

c. 𝑓

− 1

𝑥

3 +𝑥

b. 𝑓

− 1

3 −𝑥

𝑥

d. 𝑓

− 1

𝑥

3 −𝑥

9. What is the domain of f - 1

?

a. {𝑥 ≠ 0 } c. {𝑥 ≠ − 3 }

b. {𝑥 ≠ 3 } d. {𝑥 ≠ − 1 }

10. What is the range of f - 1

a.

c.

b.

d.

For numbers 11-15, consider the function 𝑓

2

11. What is the domain of the function?

a.

c.

b. {𝑥 Є ≠ 2 } d. {𝑥 Є 𝑅}

12. What is the range of the function?

a.

c.

b.

d.

Lesson

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

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 𝑓

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

we get 𝑥 = 3 𝑦 − 8.

Solving y for x: 3 𝑦 = 𝑥 + 8

𝑥+ 8

3

Therefore, the inverse of 𝑓

= 3 𝑥 − 8 is 𝑓

− 1

𝑥+ 8

3

To verify if f and f

• 1

are inverse functions:

𝑓[𝑓

− 1

(𝑥)] = 3 (

𝑥+ 8

3

− 1

[𝑓(𝑥)] =

3 𝑥− 8 + 8

3

= x+8 =

3 𝑥

3

=x = x

Therefore, f

- 1

is the inverse of f.

Example 2: Find the inverse of 𝑓

Solution: The equation of a function is 𝑦 = √

2 𝑥 + 1. Interchanging the x and y variables,

we get 𝑥 = √

Solving y for x: 2 𝑦 = 𝑥

2

𝑥

2

− 1

2

Therefore, the inverse of 𝑓

= √ 2 𝑥 + 1 is 𝑓

− 1

𝑥

2

− 1

2

To verify if f and f

• 1

are inverse functions:

[

− 1

)]

𝑥

2

− 1

2

− 1

[

]

(√ 2 𝑥+ 1 )

2

− 1

2

2

2 𝑥+ 1 − 1

2

=x = x

Therefore, f

- 1

is the inverse of f.

Example 3: Find the inverse of 𝑓

2

Solution: The equation of a function is 𝑦 = 𝑥

2

• 4. Interchanging the x and y variables,

we get 𝑥 = 𝑦

2

Solving y for x: 𝑦

2

Therefore, the inverse of 𝑓

2

• 4 is 𝑓

− 1

To verify if f and f

• 1

are inverse functions:

[

− 1

)]

2

− 1

[

]

2

2

=x = x

Therefore, f

- 1

is the inverse of f.

Notes to the Teacher

The notation f

- 1

is used to represent the inverse of a function f.

To verify that the f and f

- 1

are inverse functions:

and

C. Answer the following questions:

1. What can you say about the two given functions?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

_______________________________________________

2. Based from the table of values, describe the domain and range of the first

function with respect to the domain and range of the other function.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

_______________________________________________

3. What can you say about the graphs of the two functions?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

_______________________________________________

4. Drawing a diagonal line (y=x), what can you say about the graphs with

respect to line y=x?

_______________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

___________________________________________________

5. Can you give any other observation/s?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

_________________________________________________________

What is It

In the activity that you have done, were you able to

determine the relationship of the domain and range of the function and its inverse? Have

you seen their graphs? You will find out the easy way and understand it clearly as you

go through the next session of this module.

From the previous lesson, you already learned that the domain of a function is the set

of input values that are used for the independent variable and the range of a function

is the set of output values for the dependent variable. But, from this lesson, how will

you determine the domain and range of an inverse function?

A relation reversing the process performed by any function f(x) is called inverse of f(x).

To determine the domain and range of an inverse function:

The outputs of the function f are the inputs to f

− 1

, so the range of f is also the domain

of f

− 1

. Likewise, because the inputs to f are the outputs of f

− 1

, the domain of f is the

range of f

− 1

. We can visualize the situation.

This means 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.

f(x)

Domain of f Range of f

Range of f

• 1 Domain of f - 1

f

- 1

(x)

a b

To verify if f and f

• 1

are inverse functions:

[

− 1

)]

− 1

[

]

= x-12+12 = 3x/

=x = x

Therefore, f

- 1

is the inverse of f.

Example 3. Find the domain and range of 𝒇(𝒙) = √

𝒙 + 𝟐 and its inverse.

Solution:

Let 𝒚 = √

Interchange x and y: 𝒙 = √

Solve for y.

𝟐

𝟐

−𝟏

𝟐

Determine the domain and range of f and f

- 1

.

You have 𝒇

= √𝒙 + 𝟐 a and 𝒇

−𝟏

𝟐

Domain of (f) ={𝒙 ≥ −𝟐} Range of (f)= {𝒚 ≥ 𝟎}

Domain of (f

• 1

Range of (f

• 1

To verify if f and f

• 1

are inverse functions:

𝑓[𝑓

− 1

] = √𝑥

2

− 1

[𝑓(𝑥)] = (

2

2

= x+2- 2

=x = x

Therefore, f

- 1

is the inverse of f.

Example 4. Consider f

𝟐

− 𝟓. Find the inverse and its domain and range.

Solution:

Let 𝒚 = 𝒙

𝟐

Interchange x and y: 𝒙 = 𝒚

𝟐

Solve for y.

𝟐

−𝟏

Determine the domain and range of f and f

- 1

.

You have 𝒇

𝟐

− 𝟓 and 𝒇

−𝟏

Domain of ( f ) ={𝒙 𝝐 𝑹} Range of (f)= {𝒚 > −𝟓}

Domain of ( f

- 1

) =

Range of (f

• 1

) =

To verify if f and f

• 1

are inverse functions:

− 1

[

]

2

[

− 1

)]

2

= x+5- 5 = √𝑥

2

=x = x

Therefore, f

- 1

is the inverse of f.

What’s More

Practice Activity

A. Find the inverse of f. Determine the domain and range of each resulting inverse

functions. Write your answer inside the box provided.

f

- 1 =

Solution:

Domain

Range

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 expressed in degrees Celcius or degress 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 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 not correctly

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 c. Real Numbers

b.Domain d. Inverse Function

3. Which of the following is not allowed as the value of the independent variable

under the square root sign?

a. zero c. decimal number

b. negative number d. fraction

3. What is the domain and range of the function

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

a. 𝑓

− 1

𝑥− 5

9

c. 𝑓

− 1

𝑥+ 5

3

b. 𝑓

− 1

9

𝑥− 5

d. 𝑓

− 1

9

𝑥+ 5

5. Which of the following pair of functions is NOT the inverse of each other?

a. 𝑓

𝑥

5

b.𝑓

2 −𝑥

3

c. 𝑓

1

𝑥

1

𝑥

d.𝑓

2

For numbers 6-10, consider the function 𝑓(𝑥)

3

𝑥− 2

6. What is the domain of the function?

a. {𝑥 ≠ 3 } c. {𝑥 ≠ 2 }

b.

d.

7. What is the Range of the function?

a.

c.

d.

8. What is the inverse of the function?

− 1

2 𝑥− 3

𝑥

c. 𝑓

− 1

𝑥

2 𝑥− 3

b. 𝑓

− 1

2 𝑥+ 3

𝑥

d. 𝑓

− 1

𝑥

2 𝑥+ 3

9. What is the domain of f

- 1

𝑎. {𝑥 ≠ 0 } c. {𝑥 ≠ − 2 }

𝑏. {𝑥 ≠ 2 } d. {𝑥 ≠ 3 }

10.What is the range of f

- 1

?

a.

c.

b. {𝑦 ≠ − 3 } d. {𝑦 ≠ 2 }

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