Relasi dan Fungsi [Part 2] - Konsep Fungsi (Domain, Kodomain, Range)

Introduction

This tutorial provides a comprehensive overview of the concepts related to functions, specifically focusing on domain, codomain, and range, as discussed in the video "Relasi dan Fungsi [Part 2] - Konsep Fungsi" by Benni Al Azhri. Understanding these concepts is crucial for mastering mathematical functions, especially for students in 8th grade.

Step 1: Understanding Functions

• Definition of a Function: A function is a specific type of relation where each input (or domain) is associated with exactly one output (or codomain).
• Key Characteristics:
  • Each element in the domain maps to one and only one element in the codomain.
  • Functions can be represented as equations, graphs, or tables.

Step 2: Differentiating Between Relations and Functions

• Relation: A relation is any set of ordered pairs. It does not require that each input maps to a unique output.
• Function: A function is a special type of relation. To determine if a relation is a function:
  • Use the Vertical Line Test on the graph. If a vertical line crosses the graph more than once, it is not a function.

Step 3: Exploring Domain, Codomain, and Range

• Domain: The set of all possible input values (x-values) for a function.
  • Example: For the function f(x) = x^2, the domain is all real numbers.
• Codomain: The set of all possible output values (y-values) that the function could possibly produce.
  • Example: For the function f(x) = x^2, the codomain can be all non-negative real numbers.
• Range: The actual set of output values produced by the function from the given domain.
  • Example: For f(x) = x^2 when the domain is all real numbers, the range is all non-negative real numbers.

Step 4: Practical Examples

• Example 1: Consider the function f(x) = 2x + 3.

  • Domain: All real numbers.
  • Codomain: All real numbers.
  • Range: All real numbers since the function can produce all values as x varies.

• Example 2: For f(x) = sqrt(x).

  • Domain: x ≥ 0 (only non-negative numbers, since you can't take the square root of negative numbers).
  • Codomain: All real numbers.
  • Range: y ≥ 0 (actual outputs will only be non-negative).

Step 5: Solving Problems

• Practice Problems:
  • Identify the domain, codomain, and range for the following functions:
    1. f(x) = 1/x
    2. f(x) = x^2 - 4
  • Solutions:
    1. For f(x) = 1/x:
       • Domain: x ≠ 0 (cannot divide by zero)
       • Codomain: All real numbers
       • Range: All real numbers except 0
    2. For f(x) = x^2 - 4:
       • Domain: All real numbers
       • Codomain: All real numbers
       • Range: y ≥ -4

Conclusion

In summary, this guide covers the fundamental concepts of functions, including their definitions, differences from relations, and the critical elements of domain, codomain, and range. Understanding these concepts is key to solving mathematical problems related to functions. For further practice, consider exploring more examples and problems to reinforce your understanding. Keep studying and apply these concepts to enhance your mathematical skills!