General Mathematics Quarter 1 Module 20: Domain and Range of Exponential Functions

General Mathematics Quarter 1 – Module 20: Domain and Range of Exponential Functions

Senior High School

General Mathematics Quarter 1 – Module 20: Domain and Range of Exponential Functions

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

CO_Q1_General Mathematics SHS Module 20

General Mathematics Alternative Delivery Mode Quarter 1 – Module 20: Domain and Range of Exponential Functions First Edition, 2021

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

Development Team of the Module

Writer: Dennis E. Ibarrola

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General Mathematics Quarter 1 – Module 20: Domain and Range of Exponential Functions

Senior High School

[transcriber’s note: this page repeats the cover title and is kept as printed for the document’s title page layout]

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.

[illustration: hand pointing with a finger, inside a circle]

What I Need to Know

In this learning module, you will know more about the domain and range. These two concepts are always included in the study of functions. Identifying the domain of the function will help you to determine if the function will exist in a particular value. The domain of a function is always associated with the possible inputs for a function to exist, while the range is associated with outputs after substituting the possible inputs to the unknown variable.

This module was designed and written with you in mind. It is hoped to answer the questions, “Why domain and range are important?” and “How can I determine the domain and range of an exponential function?”. It is here to help you master finding the domain and range of an exponential function.

After going through this module, you are expected to:

1. define domain and range;
2. find the domain and range of a given function; and
3. represent the domain and range using the set builder and interval notation.

What I Know

[illustration: hand holding a pencil, inside a circle]

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

1. The set of all allowable values of the independent variable is called _____.
   a. Range
   b. Domain
   c. Real Numbers
   d. Exponential Function

   
   

2. Which of the following does not exist?
   a. Zero value on the denominator
   b. Positive values under the square root sign
   c. Negative values on the denominator
   d. Zero value under the square root sign

   
   

3. What is the domain and range of the function: (𝑥) = 3/(𝑥+7)?
   a. The domain is all real numbers except -7 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 -3 and the range is all real numbers.
   d. The domain and range are all real numbers.

   
   

4. Which of the following statement is always true?
   a. For any given x-value, the y-value of 𝑦 = 2^𝑥 is negative.
   b. For any given x-value, the y-value of 𝑦 = 2^𝑥 is all real numbers.
   c. The domain of exponential function 𝑦 = 2^𝑥 is positive numbers.
   d. The domain of exponential function 𝑦 = 2^𝑥 is all real numbers.

   
   

5. What is the domain of the function: 𝑓(𝑥) = 3^𝑥?
   a. All negative numbers
   b. All positive numbers
   c. All real numbers except 0
   d. All real numbers

   
   

For item numbers 6-7, refer to the graph below:

[chart: line graph of an increasing exponential curve; no number and title printed. Horizontal axis labels: -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. Vertical axis labels: -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. Values are not labeled on the chart.]

6. What is the domain of the exponential function?
   a. {𝑥|𝑥 ∈ ℝ, 𝑥 < 0}
   b. {𝑥|𝑥 ∈ ℝ, 𝑥 ≥ 0}
   c. {𝑥|𝑥 ∈ ℝ, 𝑥 ≤ 0}
   d. {𝑥|𝑥 ∈ ℝ}

   
   

7. What is the range of the exponential function?
   a. {𝑦|𝑦 ∈ ℝ}
   b. {𝑦|𝑦 ∈ ℝ, 𝑦 < 0}
   c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}
   d. {𝑦|𝑦 ∈ ℝ, 𝑦 ≥ 0}

   
   

For item numbers 8-9, refer to the given function: (𝑥) = 4^𝑥.

8. What is the domain of the exponential function?
   a. {𝑥|𝑥 ∈ ℝ}
   b. {𝑥|𝑥 ∈ ℝ, 𝑥 ≥ 0}
   c. {𝑥|𝑥 ∈ ℝ, 𝑥 ≤ 0}
   d. {𝑥|𝑥 ∈ ℝ, 𝑥 > 0}

   
   

9. What is the range of the exponential function?
   a. {𝑦|𝑦 ∈ ℝ}
   b. {𝑦|𝑦 ∈ ℝ, 𝑦 ≥ 0}
   c. {𝑦|𝑦 ∈ ℝ, 𝑦 ≤ 0}
   d. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}

   
   

For item numbers 10-11, refer to the given the function: 𝑓(𝑥) = (1/4)^𝑥.

10. What is the domain of the exponential function?
    a. {𝑥|𝑥 ∈ ℝ}
    b. {𝑥|𝑥 ∈ ℝ, 𝑥 > 1/4}
    c. {𝑥|𝑥 ∈ ℝ, 𝑥 > 0}
    d. {𝑥|𝑥 ∈ ℝ, 𝑥 < 1/4}

    
    

11. What is the range of the exponential function?
    a. {𝑦|𝑦 ∈ ℝ}
    b. {𝑦|𝑦 ∈ ℝ, 𝑦 > 1/4}
    c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}
    d. {𝑦|𝑦 ∈ ℝ, 𝑦 < 0}

    
    

For item numbers 12-13, refer to the given the function: 𝑓(𝑥) = 3^𝑥 + 2.

12. What is the domain of the exponential function?
    a. {𝑥|𝑥 ∈ ℝ}
    b. {𝑥|𝑥 ∈ ℝ, 𝑥 > 2}
    c. {𝑥|𝑥 ∈ ℝ, 𝑥 > 0}
    d. {𝑥|𝑥 ∈ ℝ, 𝑥 < 3}

    
    

13. What is the range of the exponential function?
    a. {𝑦|𝑦 ∈ ℝ}
    b. {𝑦|𝑦 ∈ ℝ, 𝑦 > 2}
    c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}
    d. {𝑦|𝑦 ∈ ℝ, 𝑦 < 3}

    
    

For item numbers 14-15, refer to the given the function: 𝑓(𝑥) = −3 · 2^𝑥 + 4.

14. What is the domain of the exponential function?
    a. {𝑥|𝑥 ∈ ℝ}
    b. {𝑥|𝑥 ∈ ℝ, 𝑥 > 4}
    c. {𝑥|𝑥 ∈ ℝ, 𝑥 < 3}
    d. {𝑥|𝑥 ∈ ℝ, 𝑥 < 2}

    
    

15. What is the range of the exponential function?
    a. {𝑦|𝑦 ∈ ℝ}
    b. {𝑦|𝑦 ∈ ℝ, 𝑦 > −3}
    c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 2}
    d. {𝑦|𝑦 ∈ ℝ, 𝑦 < 4}

    
    

Lesson 1: Domain and Range of Exponential Functions

For you to begin this module, you need to assess your knowledge of the basic skills in finding the domain and range of a function. From your previous lessons in Math, you already encountered the domain and range of functions. Your acquired knowledge and skill will help you understand easily how to find the domain and range of an exponential function. Seek the assistance of your teacher if you encounter any difficulty.

What’s In

[illustration: hands holding puzzle pieces]

Recall that the domain of a function is the set of all allowable values of 𝑥, commonly known as the independent variable or possible inputs of the function.

The range of a function is the set of output values commonly known as the dependent variable when all x-values in the domain are evaluated into the function.

This means that you need to find the domain first to describe the range. The following will help you to recall how to find the domain and range of a function.

Find the domain and range of the following:

1. 𝑦 = 2𝑥 + 3

Solution:

In a linear function, any real number can be substituted to 𝑥 to get an output. Therefore the domain and range are all real numbers. You can express your answer in set notation or interval notation.

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 ∈ ℝ }
Interval Notation	(−∞, +∞)	(−∞, +∞)

2. 𝑦 = 𝑥² + 2

Solution:

This is a quadratic function, like the linear function any real number can be substituted to x to get an output. You also learned that the graph of a quadratic function is a parabola that opens upward or downward. Therefore, it has a minimum or a maximum point called the vertex of the parabola.

The given 𝑦 = 𝑥² + 2 is a parabola opening upward with vertex at (0, 2), and so the lowest possible value of the function is 2. That means that the range of the function is all real numbers greater than or equal to 2.

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 𝜖 ℝ, 𝑦 ≥ 2}
Interval Notation	(−∞, +∞)	[2, +∞)

3. 𝑦 = 2/(𝑥+3)

Solution:

This is a rational function. The domain of a rational function is restricted at the value of its denominator. The denominator of a rational function should not be equal to zero for the value of the function to exist. In this case, the denominator is 𝑥 + 3, and when 𝑥 = −3 you will get a value of zero. Therefore, the domain of the function is all real numbers except −3.

Because the function 𝑦 = 2/(𝑥+3) will never be zero, you need to exclude 0 from the range. That means the range is all real numbers except 0.

	Domain	Range
Set Notation	{𝑥|𝑥 𝜖 ℝ, 𝑥 ≠ −3 }	{𝑦|𝑦 𝜖 ℝ, 𝑦 ≠ 0 }
Interval Notation	(−∞, −3) ∪ (−3, +∞)	(−∞, 0) ∪ (0, +∞)

4. 𝑦 = √(𝑥 + 5)

Solution:

The given is a radical function also known as the square root function. The domain of a radical function is any 𝑥 value for which the radicand (the value inside the radical symbol) is not negative. If the radicand has a negative value the roots or the solution are imaginary roots or no real roots. Thus, it is not allowed in the domain of the function to have a negative value inside the radical sign. Since inside the radical symbol is 𝑥 + 5 the domain of the function is a set of all possible values which are greater than or equal to -5.

Likewise, the value of the function considers positive roots only therefor the range is the set of real numbers greater than or equal to 0.

	Domain	Range
Set Notation	{𝑥|𝑥 𝜖 ℝ, 𝑥 ≥ −5 }	{𝑦|𝑦 ≥ 0 }
Interval Notation	[−5, +∞)	(0, +∞)

Notes to the Teacher

Remind the students that to find the domain of the functions they need to avoid zero (0) value in the denominator of a fraction or negative values inside the square root sign. Moreover, do mention to them that the domain of the polynomial functions which includes linear, quadratic, or any polynomial function of degree 𝑛 is always the set of real numbers.

Tell them also that the range of a function is the spread of possible y-values (minimum y-value to maximum y-value) and it can be computed by substituting different x-values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?). Finally, make sure you look for minimum and maximum values of y because it is very important to find the range of the function.

Can You Show Me the Way?

What’s New

Complete the table of values of the exponential function. Then, draw its graph in the given coordinate plane.

x	-3	-2	-1	0	1	2	3
𝑓(𝑥) = 3^𝑥

[chart: blank coordinate plane with x- and y-axes and grid lines; no values printed]

[illustration: hand holding a sprouting plant]

1. How will you describe the graph?

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2. In which quadrant/s do the graph occupies?

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3. What are the other possible values of x? Describe the range of the possible values of x?

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4. What do you think now is the range of the function?

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5. What other observations can you give?

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What is It

[illustration: magnifying glass held by a hand]

In the activity that you have done, were you able to determine the possible values of x? How about the range of the given function? You will find whether your answers are correct as you go through the next session of this module.

The Domain and Range

The domain of a function is the set of input values that are used for the independent variable. The range of a function is the set of output values for the dependent variable. For any exponential function, 𝑓(𝑥) = 𝑏^𝑥 the domain is the set of all real numbers. The range, however, is bounded by the horizontal asymptote of the graph of 𝑓(𝑥).

Example 1: Find the domain and range of the function 𝑦 = 3^(𝑥+2)

Solution:

Look at the graph of the function.

[chart: graph of the exponential function in Example 1, rising from left to right, with the x-axis labeled from -9 to 3 and the y-axis labeled from -1 to 6; no values otherwise printed]

The function is defined for all real numbers. So, the domain of the function is a set of real numbers. As 𝑥 extends to approach positive infinity (+∞), the value of the function also extends to +∞, and as 𝑥 extends to approach negative infinity (−∞), the function approaches the x-axis but never touches it. Therefore, the range of the function is a set of real positive numbers greater than 0 or {𝑦|𝑦 𝜖 ℝ, 𝑦 > 0 }.

Thus, the domain and range of the given function is given below and can be written as:

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 𝜖 ℝ, 𝑦 > 0 }
Interval Notation	(−∞, +∞)	(0, +∞)

Example 2: Find the domain and range of the function 𝑦 = (1/4)^(2𝑥)

Solution:

Look at the graph on the next page in the coordinate plane.

[chart: graph of the exponential function in Example 2, falling from left to right and approaching the x-axis, with a marked point at (0, 1); axes marked from -5 to 8 horizontally and -1 to 6 vertically]

The function is defined for all real numbers. So, the domain of the function is the set of real numbers.

As x tends to approach +∞, the value of the function tends to approach zero, and the graph approaches the x-axis but never touches it. As x tends to approach −∞, the function also tends to −∞. Therefore, the range of the function is the set of real positive numbers greater than 0 or {𝑦|𝑦 𝜖 ℝ, 𝑦 > 0 }.

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 𝜖 ℝ, 𝑦 > 0 }
Interval Notation	(−∞, +∞)	(0, +∞)

Domain and Range of Exponential Functions

Let f(x) = 𝑎 · 𝑏^(𝑃(𝑥)) + ℎ be an exponential function where 𝑃(𝑥) is linear. Then

Then, the domain of the function is ℝ

Range of the function, =

• (ℎ, +∞), if 𝑎 > 0
• (−∞, ℎ), if 𝑎 < 0

In cases of exponential functions where 𝑃(𝑥) is linear, in which case, 𝑏^(𝑝(𝑥)) will always be defined for any value of 𝑥. Thus, the domain of an exponential function is the set of real numbers or ℝ. For the range, note that 𝑏^(𝑝(𝑥)) > 0 for any values of 𝑥. Hence, the range of an exponential function will depend on 𝑎 and ℎ.

Example 3: Let 𝑓(𝑥) = 3^𝑥. Find the domain and range.

Solution:

The domain of the function is the set of real numbers since 𝑓(𝑥) = 3^𝑥 is defined for any real number x. It means that any value of 𝑥 from the set of real numbers can be substituted to variable 𝑥. Note that any power of 3 is always positive. Hence, the range is (0, +∞).

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 𝜖 ℝ, 𝑦 > 0 }
Interval Notation	(−∞, +∞)	(0, +∞)

Example 4: Let 𝑓(𝑥) = 4^(𝑥+1) − 2. Find the domain and range.

Solution:

The domain of the function is the set of real numbers because 𝑃(𝑥) = 𝑥 + 1 and it is linear. Also, in the given function you may observe that 𝑎 > 0 (𝑎 = 1 𝑎𝑛𝑑 𝑏 = 4) and ℎ = −2, hence the range of the function is equal to (ℎ, +∞).

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 𝜖 ℝ, 𝑦 > −2 }
Interval Notation	(−∞, +∞)	(−2, +∞)

Example 5. Let 𝑓(𝑥) = −2^(𝑥−1) + 3. Find the domain and range.

Solution:

The domain of the function is the set of real numbers because 𝑃(𝑥) = 𝑥 − 1 and it is linear. Also, in the given function you may observe that 𝑎 < 0 (𝑎 = −1 𝑎𝑛𝑑 𝑏 = 2) and ℎ = 3, hence the range of the function is equal to (−∞, ℎ).

	Domain	Range
Set Notation	{𝑥|𝑥 ∈ ℝ }	{𝑦|𝑦 𝜖 ℝ, 𝑦 < 3 }
Interval Notation	(−∞, +∞)	(−∞, 3)

What’s More

Activity 1.1

Answer the guide questions to complete the table of domain and range of the following exponential functions.

a. 𝑓(𝑥) = 5^𝑥

Is 𝑓(𝑥) = 5^𝑥 defined at any values of 𝑥? _____________________________.

What is the minimum value of 𝑓(𝑥)? _____________________________.

Can you determine the the maximum value of 𝑓(𝑥)? _____________.

	Domain	Range
Set Notation
Interval Notation

b. 𝑓(𝑥) = (1/5)^3𝑥

Is 𝑓(𝑥) = (1/5)^3𝑥 defined at any values of 𝑥? _____________________.

What is the minimum value of 𝑓(𝑥)? _____________________________.

Can you determine the maximum value of 𝑓(𝑥)? _________________.

	Domain	Range
Set Notation
Interval Notation

c. 𝑓(𝑥) = 1/3 (2)^𝑥 + 3

Which is 𝑃(𝑥)? _______________Is 𝑃(𝑥) linear? _________________

Is 𝑎 > 0 or 𝑎 < 0? _______________

What is the value of ℎ? ___________________

	Domain	Range
Set Notation
Interval Notation

d. 𝑔(𝑥) = −(4)^{2𝑥−3} +1

Which is 𝑃(𝑥)? _______________Is 𝑃(𝑥) linear? _________________

Is 𝑎 > 0 or 𝑎 < 0? _______________

What is the value of ℎ? ___________________

	Domain	Range
Set Notation
Interval Notation

e. ℎ(𝑥) = 2(3)^{3𝑥−1} + 2

Which is 𝑃(𝑥)? _______________Is 𝑃(𝑥) linear? _________________

Is 𝑎 > 0 or 𝑎 < 0? _______________

What is the value of ℎ? ___________________

	Domain	Range
Set Notation
Interval Notation

Activity 1.2

Find the domain and range of the following exponential functions and express your answer in set notation for numbers 1 to 5 and interval notation for numbers 6 to 10:

1. 𝑓(𝑥) = (1/2)^{𝑥+1}
2. 𝑓(𝑥) = 3^{𝑥+2}
3. 𝑓(𝑥) = −(5)^𝑥
4. 𝑓(𝑥) = 4^{−𝑥}
5. 𝑓(𝑥) = (1/3)^{−𝑥+1}
6. 𝑓(𝑥) = 2^{𝑥−1} + 1
7. 𝑓(𝑥) = −5^𝑥 − 2
8. 𝑓(𝑥) = (1/4)^{2𝑥+1} + 3
9. 𝑓(𝑥) = 5^{3𝑥} − 4
10. 𝑓(𝑥) = −(2/3)^𝑥 − 3

What I Have Learned

Warm that Mind Up!

A. Fill in the blanks with the correct word or phrase to complete the sentence.

1. The set of values of the independent variable (usually 𝑥) is called _________.
2. The resulting 𝑦-values after we have substituted in the possible 𝑥-values is called _____________.
3. In the function 𝑓(𝑥) = 𝑎 ∙ 𝑏^{𝑃(𝑥)} + ℎ where 𝑃(𝑥) is linear, the domain of the function is _____________.
4. In the function 𝑓(𝑥) = 𝑎 ∙ 𝑏^{𝑃(𝑥)} + ℎ where 𝑃(𝑥) is linear, if 𝑎 > 0 the range of the function is ___________.
5. In the function 𝑓(𝑥) = 𝑎 ∙ 𝑏^{𝑃(𝑥)} + ℎ where 𝑃(𝑥) is linear, if 𝑎 < 0 the range of the function is ____________.

B. In your own words, what technique or strategy can you think of to facilitate your way of finding the domain and range of an exponential function?

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C. This lesson helps you understand the procedure in finding the domain and range of an exponential function. You also discover your own technique or strategy on how to easily find ways in determining the domain and range of an exponential function. If you have given the chance to give an advice to Jethro who is a Senior High School learner that is working as a part time at the canteen near his house at the same time studying, what strategy or technique can you suggest to finish his study. Explain

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

Mission Possible

What I Can Do

1. The volume V of air remaining in an inflated balloon can be modeled by the function 𝑉 = 1,000(0.85)^𝑥 where x represents the number of days that have passed since inflating the balloon. What is the reasonable domain for the situation? Explain.

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2. The function 𝑓(𝑥) = 65,000(1.5)^𝑥 can be modeled the population of a city for x, the number of years that have passed since 2010. What inequality represents the reasonable range of the function based on the situation? Explain.

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Rubrics:

5- Shows in-depth comprehension of the pertinent concepts and /or processes, and provides explanations wherever appropriate.

4- Shows in-depth comprehension of the pertinent concepts and/or processes.

3- Shows in-depth comprehension of major concepts although neglects or misinterprets less significant ideas or details.

2- Shows gaps in theoretical comprehension.

1- Demonstrate minor comprehension not being able to develop an approach.

Assessment

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

1. The resulting y-values after we have substituted in the possible x-values is called _________.

   
   

   a. Range

   
   

   b. Domain

   
   

   c. Real Numbers

   
   

   d. Exponential Function

   
   

2. Which of the following does not exist?

   
   

   a. zero on the numerator

   
   

   b. zero values under the square root sign

   
   

   c. negative values on the denominator

   
   

   d. negative values under the square root sign

   
   

3. What is the domain and range of the function: f(𝑥) = √(𝑥 + 7) ?

   
   

   a. Both the domain and the range are a set of all real numbers.

   
   

   b. The domain is the set of all real numbers while the range is a set of real numbers greater than or equal to 0.

   
   

   c. The domain is a set of real numbers greater than or equal to -7 while the range is a set of real numbers all real numbers.

   
   

   d. The domain is a set of real numbers greater than or equal to -7 while the range is a set of real numbers greater than or equal to 0.

   
   

4. Which of the following statement is never true?

   
   

   a. For any given x-value, the y-value of 𝑦 = 5^𝑥 is negative.

   
   

   b. For any given x-value, the y-value of 𝑦 = 5^𝑥 is positive.

   
   

   c. The domain of an exponential function 𝑦 = 5^𝑥 includes positive numbers.

   
   

   d. The domain of an exponential function 𝑦 = 5^𝑥 is all real numbers.

   
   

5. Which of the following statements is true about the function 𝑦 = 3^𝑥?

   
   

   a. The range of the function is the set of all real numbers.

   
   

   b. The domain of the function is the set of real numbers less than 3.

   
   

   c. The range of the function is the set of all real numbers less than 3.

   
   

   d. The domain of the function is the set of all real numbers.

   
   

Use the graph below for nos. 6-7.

[chart: line graph of a decreasing exponential curve; no title printed; x-axis labeled with numbers from -6 to 8; y-axis labeled with numbers from 1 to 6]

Values are not labeled on the chart.

6. What is the domain of the exponential function 𝑓(𝑥) = 𝑏^𝑥?

   
   

   a. {𝑥|𝑥 ∈ ℝ, 𝑥 ≥ 0}

   
   

   c. {𝑥|𝑥 ∈ ℝ }

   
   

   b. {𝑥|𝑥 ∈ ℝ, 𝑥 < 0}

   
   

   d. {𝑥|𝑥 ∈ ℝ, 𝑥 ≤ 0}

   
   

7. What is the range of the exponential function 𝑓(𝑥) = 𝑏^𝑥?

   
   

   a. {𝑦|𝑦 ∈ ℝ }

   
   

   c. {𝑦|𝑦 ∈ ℝ , y < 0}

   
   

   b. {𝑦|𝑦 ∈ ℝ , y ≥ 0}

   
   

   d. {𝑦|𝑦 ∈ ℝ , y > 0}

   
   

For item numbers 8-9, refer to the given function 𝑓(𝑥) = 3^{𝑥+1}

8. What is the domain of the exponential function?

   
   

   a. {𝑥|𝑥 ∈ ℝ }

   
   

   c. {𝑥|𝑥 ∈ ℝ, 𝑥 ≤ 0}

   
   

   b. {𝑥|𝑥 ∈ ℝ, 𝑥 ≥ 0}

   
   

   d. {𝑥|𝑥 ∈ ℝ, 𝑥 > 0}

   
   

9. What is the range of the exponential function?

   
   

   a. {𝑦|𝑦 ∈ ℝ }

   
   

   c. {𝑦|𝑦 ≤ 0}

   
   

   b. {𝑦|𝑦 ∈ ℝ, 𝑦 ≥ 0}

   
   

   d. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}

   
   

For item numbers 10-11, refer to the given function 𝑓(𝑥) = (1/3)^𝑥.

10. What is the domain of the exponential function?

    
    

    a. {𝑥|𝑥 ∈ ℝ }

    
    

    c. {𝑥|𝑥 ∈ ℝ, 𝑥 > 0}

    
    

    b. {𝑥|𝑥 ∈ ℝ, 𝑥 > 1/3}

    
    

    d. {𝑥|𝑥 ∈ ℝ, 𝑥 < 1/3}

    
    

11. What is the range of the exponential function?

    
    

    a. {𝑦|𝑦 ∈ ℝ }

    
    

    c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}

    
    

    b. {𝑦|𝑦 ∈ ℝ, 𝑦 > 1/3}

    
    

    d. {𝑦|𝑦 ∈ ℝ, 𝑦 < 0}

    
    

For item numbers 12-13, refer to the given function 𝑓(𝑥) = 5^𝑥 + 2

12. What is the domain of the exponential function?

    
    

    a. {𝑥|𝑥 ∈ ℝ }

    
    

    c. {𝑥|𝑥 ∈ ℝ, 𝑥 > 0}

    
    

    b. {𝑥|𝑥 ∈ ℝ, 𝑥 > 2}

    
    

    d. {𝑥|𝑥 ∈ ℝ, 𝑥 < 5}

    
    

13. What is the range of the exponential function?

    
    

    a. {𝑦|𝑦 ∈ ℝ }

    
    

    c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 0}

    
    

    b. {𝑦|𝑦 ∈ ℝ, 𝑦 > 2}

    
    

    d. {𝑦|𝑦 ∈ ℝ, 𝑦 < 2}

    
    

For item numbers 14-15, refer to the given function 𝑓(𝑥) = −4(3)^𝑥 − 5.

14. What is the domain of the exponential function?

    
    

    a. {𝑥|𝑥 ∈ ℝ }

    
    

    c. {𝑥|𝑥 ∈ ℝ, 𝑥 < 3}

    
    

    b. {𝑥|𝑥 ∈ ℝ, 𝑥 > −4}

    
    

    d. {𝑥|𝑥 ∈ ℝ, 𝑥 < −5}

    
    

15. What is the range of the exponential function?

    
    

    a. {𝑦|𝑦 ∈ ℝ }

    
    

    c. {𝑦|𝑦 ∈ ℝ, 𝑦 > 3}

    
    

    b. {𝑦|𝑦 ∈ ℝ, 𝑦 > −4}

    
    

    d. {𝑦|𝑦 ∈ ℝ, 𝑦 < −5}

    
    

Let me Formulate!

Additional Activities

In this section, you are going to think deeper and test further your understanding of the domain and range of the exponential function.

Give five examples of exponential functions in the form 𝑓(𝑥) = 𝑏^𝑥 and 𝑓(𝑥) = 𝑎 ∙ 𝑏^{𝑃(𝑥)} + ℎ with its domain and range.

Answer Key

[transcriber’s note: this page is printed upside down; the answers below were read from the rotated image]

What I Know

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

Assessment

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

What’s More

Activity 1.1

	Range	Domain
Set Notation	{y | y > 0}	{x | x ∈ ℝ}
Interval Notation	(0, +∞)	(−∞, +∞)

	Range	Domain
Set Notation	{y | y > 0}	{x | x ∈ ℝ}
Interval Notation	(0, +∞)	(−∞, +∞)

	Range	Domain
Set Notation	{y | y > 3}	{x | x ∈ ℝ}
Interval Notation	(3, +∞)	(−∞, +∞)

	Range	Domain
Set Notation	{y | y < 1}	{x | x ∈ ℝ}
Interval Notation	(−∞, 1)	(−∞, +∞)

	Range	Domain
Set Notation	{y | y > 2}	{x | x ∈ ℝ}
Interval Notation	(2, +∞)	(−∞, +∞)

Activity 1.2

1. D: {x | x ∈ ℝ}; R: {y | y > 0} 6. D: (−∞, +∞); R: (1, +∞)

   
   

2. D: {x | x ∈ ℝ}; R: {y | y > 0} 7. D: (−∞, +∞); R: (−∞, −2)

   
   

3. D: {x | x ∈ ℝ}; R: {y | y < 0} 8. D: (−∞, +∞); R: (3, +∞)

   
   

4. D: {x | x ∈ ℝ}; R: {y | y > 0} 9. D: (−∞, +∞); R: (−4, +∞)

   
   

5. D: {x | x ∈ ℝ}; R: {y | y > 0} 10. D: (−∞, +∞); R: (−∞, −3)

   
   

References

Alday, Eward M., Batisan, Ronaldo S., and Caraan, Aleli M. General Mathematics. Makati City: Diwa Learning Systems Inc., 2016. 70-76, 120-130, 176-201.

Orines, Fernando B., Esparrago, Mirla S., and Reyes, Junior. Nestor V. Advanced Algebra: Trigonometry and Statistics. Second Edition. Quezon City: Phoenix Publishing House Inc., 2004. 249-253.

Oronce, Orlando A., and Mendoza, Marilyn O. General Mathematics. Quezon City: Rex Bookstore, Inc., 2016. 186-202

Oronce, Orlando. General Mathematics. Quezon City: Rex Bookstore, Inc., 2016. 107-151

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

*DepED Material: General Mathematics Learner’s Material

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