Lesson Plan | Traditional Methodology | Function: Codomain and Range

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to introduce the concepts of codomain and image of a function, providing a solid foundation for students to understand the difference between these two terms and how to calculate the image of a function. This understanding is crucial, as it allows students to recognize the relationship between the elements of the domain and the codomain, and how this relationship manifests in the image of the function.

Main Objectives

1. Differentiate between the codomain and the image of a function.

2. Calculate the image of a given function.

3. Verify when the codomain is equal to the image.

Introduction

Duration: (10 - 15 minutes)

🎯 Purpose: The purpose of this stage is to introduce the concepts of codomain and image of a function, providing a solid foundation for students to understand the difference between these two terms and how to calculate the image of a function. This understanding is crucial, as it allows students to recognize the relationship between the elements of the domain and the codomain, and how this relationship manifests in the image of the function.

Context

📘 Context: Start the lesson by explaining to students that the study of functions is fundamental to Mathematics and many other disciplines, such as Physics, Economics, and even Biology. Functions help describe and predict behaviors in different situations in the real world. For example, the function that relates time and distance traveled by a car allows us to calculate the average speed of that car. To fully understand functions, it is essential to know the concepts of codomain and image.

Curiosities

🔍 Curiosity: Did you know that functions are used to create graphs of population growth, predict market trends, and even model the weather? Understanding how elements of the domain (input) relate to elements of the codomain (output) and the image (result) is crucial for these predictions. For example, recommendation algorithms from platforms like Netflix use functions to suggest movies and series based on each user's viewing history.

Development

Duration: (45 - 50 minutes)

🎯 Purpose: The purpose of this stage is to deepen the understanding of the concepts of codomain and image, providing practical and detailed examples that help students understand the differences and similarities between these terms. Additionally, the proposed questions will allow students to apply the acquired knowledge, consolidating their learning through problem-solving.

Covered Topics

1. 📘 Definition of Codomain: Explain that the codomain of a function is the set of all possible output values, that is, the set that contains all values that the function can take. This is a theoretical concept that does not depend on the specific values that the function actually assumes. 2. 📘 Definition of Image: Detail that the image of a function is the set of all output values that the function actually takes when we apply all values from the domain. In other words, it is the set of values resulting from the operations of the function. 3. 📘 Relationship between Domain, Codomain, and Image: Explain the relationship between domain, codomain, and image, highlighting that the domain is the set of all possible inputs, the codomain is the set of all possible outputs, and the image is the set of actual outputs. Use Venn diagrams to illustrate these relationships. 4. 📘 Examples of Codomain and Image: Provide concrete examples of simple functions, such as f(x) = x^2, and identify the domain, codomain, and image. Show how, for the function f(x) = x^2 with domain in real numbers, the codomain is real numbers, but the image is only non-negative real numbers. 5. 📘 Difference between Codomain and Image: Reinforce the difference between codomain and image with additional examples. For example, use the function f(x) = 2x + 1 with domain in integers and show how the codomain is all real numbers, but the image is only odd numbers.

Classroom Questions

1. 1️⃣ Consider the function f(x) = x^2 with domain in real numbers. What is the image of this function? Is the codomain equal to the image? 2. 2️⃣ Given the function f(x) = 2x + 3 with domain in integers, what is the image? Explain whether the codomain is equal to the image or not. 3. 3️⃣ For the function f(x) = |x| (absolute value of x) with domain in real numbers, determine the image and the codomain. Are they equal?

Questions Discussion

Duration: (15 - 20 minutes)

🎯 Purpose: The purpose of this stage is to review and consolidate students' learning about codomain and image, allowing them to confirm their answers and understand the detailed explanations for each question. Additionally, the proposed questions and reflections aim to engage students in a broader discussion about the application of the learned concepts, promoting a deeper and contextualized understanding of the content.

Discussion

• 1️⃣ Question: Consider the function f(x) = x^2 with domain in real numbers. What is the image of this function? Is the codomain equal to the image? Explanation: For the function f(x) = x^2, when the domain is real numbers, any real number can be an input. The codomain, in this case, is the set of all real numbers. However, the image of this function is only the non-negative real numbers (all output values of x^2 are greater than or equal to zero). Therefore, the codomain is not equal to the image.

• 2️⃣ Question: Given the function f(x) = 2x + 3 with domain in integers, what is the image? Explain whether the codomain is equal to the image or not. Explanation: The function f(x) = 2x + 3 with domain in integers generates all odd numbers as output. The codomain of this function is the set of all real numbers, but the image is only odd numbers. Therefore, the codomain is not equal to the image.

• 3️⃣ Question: For the function f(x) = |x| (absolute value of x) with domain in real numbers, determine the image and the codomain. Are they equal? Explanation: The function f(x) = |x|, where f(x) represents the absolute value of x, has its domain in real numbers. The codomain is the set of real numbers, as there are no theoretical restrictions. However, the image of this function is the set of non-negative real numbers, as the absolute value of any number is always greater than or equal to zero. Therefore, the codomain is not equal to the image.

Student Engagement

1. 🤔 Question: How can we practically determine when the codomain is equal to the image? 2. 💡 Reflection: Why is it important to differentiate between codomain and image when solving mathematical problems? 3. 📈 Question: In what everyday situations do you think knowledge about the image of a function can be applied? 4. 🔍 Reflection: How can understanding codomain and image help in understanding graphs and functions in other disciplines, such as Physics or Economics?

Conclusion

Duration: (10 - 15 minutes)

🎯 Purpose: The purpose of this stage is to review and consolidate the main points covered in the lesson, ensuring that students fully understand the concepts of codomain and image. Additionally, this stage aims to highlight the importance and practical application of the learned content, reinforcing its value for everyday life and other areas of knowledge.

Summary

• 📘 Definition of Codomain: The codomain of a function is the set of all possible output values.
• 📘 Definition of Image: The image of a function is the set of all output values that the function actually assumes when we apply all values from the domain.
• 📘 Relationship between Domain, Codomain, and Image: The domain is the set of all possible inputs, the codomain is the set of all possible outputs, and the image is the set of actual outputs.
• 📘 Examples of Codomain and Image: Concrete examples, such as f(x) = x^2 and f(x) = 2x + 3, help to differentiate between codomain and image.
• 📘 Difference between Codomain and Image: Codomain is the theoretical set of outputs, while the image is the set of actual resulting values.

The lesson connected theory to practice by providing concrete examples of functions and demonstrating how to calculate the image and identify the codomain. Using real functions like f(x) = x^2 and f(x) = 2x + 3 helped students see how these concepts apply in practical situations and in solving mathematical problems.

Understanding codomain and image is crucial in everyday life, as these concepts are used in various fields such as Physics, Economics, and even algorithm programming for recommendations. For example, streaming platforms use functions to suggest content based on users' viewing history, demonstrating the practical relevance of this knowledge.