CARA MUDAH FUNGSI KOMPOSISI

Introduction

In this tutorial, we will explore the concept of function composition in mathematics, where two functions can be combined to create a new function. Understanding function composition is essential for solving various mathematical problems, especially in algebra. This guide will provide clear examples and steps to master the topic quickly.

Step 1: Understanding Function Composition

Function composition involves two functions, denoted as f(x) and g(x). The operation is symbolized by "o" and can result in two types of compositions:

• (g o f)(x): This means function f is inserted into function g.
• (f o g)(x): This means function g is inserted into function f.

Practical Tip

To visualize function composition, consider it like passing a value through two machines (functions) where the output of the first becomes the input for the second.

Step 2: Performing Function Composition

Let’s look at an example where we have two functions:

• f(x) = 3x - 4
• g(x) = 2x

We will calculate both compositions:

Finding (f o g)(x)

1. Substitute g(x) into f(x):
   • (f o g)(x) = f(g(x)) = f(2x)
   • = 3(2x) - 4
   • = 6x - 4

Finding (g o f)(x)

2. Substitute f(x) into g(x):
   • (g o f)(x) = g(f(x)) = g(3x - 4)
   • = 2(3x - 4)
   • = 6x - 8

Summary of Results

• (f o g)(x) = 6x - 4
• (g o f)(x) = 6x - 8

Step 3: Properties of Function Composition

Function composition has distinct properties:

• Non-Commutative: (g o f)(x) ≠ (f o g)(x)
• Associative: (f o (g o h))(x) = ((f o g) o h)(x)
• Identity Function: I(x) = x, where (f o I)(x) = f(x)

Common Pitfall

Always remember that the order in which functions are composed matters. Switching the order can lead to different results.

Step 4: Determining a Function from Composition

If you know the composition of two functions and one of the functions, you can find the other.

Example Problem

Given:

• (f o g)(x) = -4x + 4
• f(x) = 2x + 2

Steps to Find g(x):

1. Set up the equation:

   • f(g(x)) = -4x + 4
   • 2(g(x)) + 2 = -4x + 4

2. Solve for g(x):

   • 2(g(x)) = -4x + 2
   • g(x) = -2x + 1

Result

• g(x) = -2x + 1

Conclusion

Function composition is a foundational concept in algebra that allows you to combine functions in useful ways. By understanding how to perform compositions and their properties, as well as how to deduce one function from another, you can tackle many mathematical problems more effectively.

To further your understanding, practice with different functions and try finding compositions and inverses to solidify your grasp of these concepts.