Komposisi Fungsi - Matematika Wajib Kelas XI Kurikulum Merdeka

Introduction

This tutorial provides a comprehensive guide to understanding the composition of functions, an essential topic in Grade XI Mathematics as per the Kurikulum Merdeka. The tutorial will break down key concepts, examples, and applications to help you grasp the material effectively.

Step 1: Understanding Functions

• Definition of Functions: A function is a relation that assigns exactly one output for each input. It can be denoted as f(x), where x is the input and f(x) is the output.

Types of Functions

• One-to-One Function: Each output is uniquely related to one input.
• Onto Function: Every possible output is mapped by at least one input.
• Notation: Familiarize yourself with function notation and how to read it.

Step 2: Introduction to Function Composition

• Definition: The composition of functions combines two functions by using the output of one function as the input of another. It is usually written as (f ∘ g)(x) = f(g(x)).

Visualizing Composition: Consider the functions f(x) and g(x). When you compose these, you first apply g to x and then apply f to the result

• Step 1: Calculate g(x)
• Step 2: Calculate f(g(x))

Step 3: Finding the Domain of Composite Functions

• Determine the Domain: The domain of a composite function is the set of all input values (x) that can be plugged into the function.
  • Check the domain of g(x) to ensure the output falls within the domain of f(x).
• Practical Tips: Always consider the restrictions that each function may impose on the composite function.

Step 4: Examples of Function Composition

• Example 1: Let f(x) = 2x + 3 and g(x) = x^2.

  Find (f ∘ g)(x)

  1. Calculate g(x): g(x) = x^2.
  2. Substitute into f: f(g(x)) = f(x^2) = 2(x^2) + 3 = 2x^2 + 3.

• Example 2: If f(x) = x + 1 and g(x) = 3x, find (g ∘ f)(x):

  • Calculate f(x): f(x) = x + 1.
  • Substitute into g: g(f(x)) = g(x + 1) = 3(x + 1) = 3x + 3.

Step 5: Understanding Inverse Functions

• Definition: An inverse function reverses the effect of a function. If f(x) transforms x into y, then its inverse f⁻¹(y) transforms y back to x.

Finding Inverses: To find the inverse of a function

1. Replace f(x) with y.
2. Swap x and y.
3. Solve for y to find f⁻¹(x).

Conclusion

Understanding composition and inverse functions is crucial for mastering higher-level mathematics. This tutorial covered key definitions, examples, and practical tips to help solidify your knowledge. As a next step, practice composing various functions and finding their inverses to enhance your skills. For further learning, explore additional resources on functions and their properties.