Fungsi Injektif, Surjektif dan Bijektif - Matematika SMA Kelas XI Kurikulum Merdeka

Introduction

This tutorial will guide you through the concepts of injective, surjective, and bijective functions, which are fundamental in understanding the behavior of mathematical functions. These concepts are particularly relevant for high school mathematics students following the Kurikulum Merdeka in Indonesia. By the end of this guide, you will be able to differentiate between these types of functions and apply them in various mathematical contexts.

Step 1: Understanding Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (codomain).
• Each input is related to exactly one output.
• Familiarize yourself with key terms:
  • Domain: The set of all possible input values.
  • Codomain: The set of all possible output values.
  • Range: The actual outputs from the function based on the domain.

Step 2: Identifying Injective Functions

• An injective function (or one-to-one function) is defined as follows:
  • Every element in the domain maps to a unique element in the codomain.
• To check if a function is injective:
  • If f(a) = f(b), then a must equal b.
• Practical tips:
  • Graphical methods: A horizontal line test can help determine if a function is injective. If any horizontal line intersects the graph at more than one point, the function is not injective.

Step 3: Identifying Surjective Functions

• A surjective function (or onto function) is defined by:
  • Every element in the codomain is the image of at least one element from the domain.
• To check if a function is surjective:
  • Ensure that for every element in the codomain, there exists at least one element in the domain that maps to it.
• Practical tips:
  • Visualize the function: If you can cover the entire codomain with outputs from the domain, the function is surjective.

Step 4: Understanding Bijective Functions

• A bijective function is both injective and surjective:
  • It establishes a one-to-one correspondence between the domain and codomain.
• Characteristics of bijective functions:
  • Each element in the domain maps to a unique element in the codomain, and vice versa.
• Practical applications:
  • Bijective functions are important for defining inverse functions, as only bijective functions can have an inverse that is also a function.

Step 5: Practical Examples

• Consider the function f(x) = 2x:
  • This function is injective because it produces a unique output for every input.
  • It is also surjective if we consider the codomain to be the set of all real numbers.
• Example of a bijective function: f(x) = x + 3
  • This function is both injective and surjective, mapping every real number uniquely to another real number.

Conclusion

Understanding injective, surjective, and bijective functions is crucial for mastering the concepts of functions in mathematics. These definitions help in analyzing the relationships between sets and are foundational for topics like function composition and inverses. To reinforce your learning, practice identifying these types of functions with various examples and consider how they apply in real-world scenarios. Explore additional resources on function composition and inverses for a deeper understanding.