Komposisi Fungsi Part 2 - Operasi Komposisi Fungsi dan Sifat-sifatnya [ Matematika Wajib Kelas X ]

Introduction

This tutorial provides a step-by-step guide on the operations of function composition and its properties, as explained in the video "Komposisi Fungsi Part 2." Understanding function composition is essential for mastering algebra concepts in mathematics, especially for students in grade X. This guide will break down the key concepts, provide examples, and suggest practice problems to enhance your understanding.

Step 1: Understanding Function Composition

Function composition involves combining two functions to form a new function. The notation for function composition is usually expressed as (f ∘ g)(x), which means f(g(x)).

Key Points

• Definition: If you have two functions, f(x) and g(x), the composition f(g(x)) means that you first apply g to x, then apply f to the result of g.
• Order Matters: The order in which functions are composed is important; f(g(x)) is not the same as g(f(x)).

Practical Advice

• Create simple functions to practice composition. For example:
  • Let f(x) = x + 2
  • Let g(x) = 3x
  • Compute f(g(1)) and g(f(1)) to see how the results differ.

Step 2: Performing Function Composition

To perform function composition, follow these steps:

1. Identify the two functions you want to compose.
2. Substitute the inner function into the outer function.
3. Simplify the result if possible.

Example

Using the functions defined earlier:

• Calculate (f ∘ g)(x):

  • g(x) = 3x
  • f(g(x)) = f(3x) = 3x + 2

• Calculate (g ∘ f)(x):

  • f(x) = x + 2
  • g(f(x)) = g(x + 2) = 3(x + 2) = 3x + 6

Common Pitfalls

• Ensure proper substitution of the inner function into the outer function.
• Double-check if you are applying the functions in the correct order.

Step 3: Exploring Properties of Function Composition

Function composition has several important properties:

• Associativity: (f ∘ g) ∘ h = f ∘ (g ∘ h)
• Identity Function: For any function f, f ∘ I = f and I ∘ f = f, where I is the identity function defined as I(x) = x.

Practical Advice

• Test these properties with different functions to see how they hold true.
• Use simple functions initially, then progress to more complex ones.

Step 4: Practice Problems

To reinforce your understanding, try these practice problems:

1. From the following functions, calculate the compositions:

   • h(x) = x^2
   • k(x) = x - 1
   • Compute (h ∘ k)(2) and (k ∘ h)(2).

2. Prove that the composition of two identity functions is still an identity function.

Solutions

• For problem 1, calculate:
  • (h ∘ k)(2) = h(k(2)) = h(2 - 1) = h(1) = 1^2 = 1
  • (k ∘ h)(2) = k(h(2)) = k(2^2) = k(4) = 4 - 1 = 3

Conclusion

In this tutorial, you learned about function composition, how to perform it, and explored its fundamental properties. Practicing with various functions will deepen your understanding of these concepts. As a next step, consider watching the additional videos mentioned for further exploration of function inverses and more complex problems. Happy studying!