Fungsi Komposisi Dua Fungsi, Matematika SMA-SMK Kelas 11 Kurikulum Merdeka @matematika-asik

Introduction

This tutorial will guide you through the concept of function composition, specifically focusing on the composition of two functions. This topic is essential for students in high school mathematics, particularly those in the 11th grade following the Merdeka Curriculum. Understanding function composition will enhance your problem-solving skills in mathematics and prepare you for more complex topics in calculus and algebra.

Step 1: Understanding Functions

• A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.
• Notation: If f(x) is a function of x, then for every x, there is a unique f(x).

Key Points

• Example of a function: f(x) = 2x + 3.
• The domain is the set of all possible inputs (x-values), and the range is the set of all possible outputs (f(x)-values).

Step 2: Defining Function Composition

• Function composition involves combining two functions to create a new function.
• Notation: If you have two functions, f and g, the composition of f and g is denoted as (f ∘ g)(x) = f(g(x)).

Practical Advice

• To find (f ∘ g)(x), first evaluate g(x) and then substitute that result into f.

Example

• Let f(x) = x + 1 and g(x) = 2x.

To find (f ∘ g)(x)

1. Calculate g(x): g(x) = 2x.
2. Substitute g(x) into f: f(g(x)) = f(2x) = 2x + 1.
• Therefore, (f ∘ g)(x) = 2x + 1.

Step 3: Working with Specific Functions

• Practice with different types of functions to solidify your understanding of composition.

Consider functions like

• Linear functions (e.g., f(x) = x + 2).
• Quadratic functions (e.g., g(x) = x²).
• Trigonometric functions (e.g., h(x) = sin(x)).

Example Problems

1. Given f(x) = x² and g(x) = x - 3, find (f ∘ g)(x):

   • g(x) = x - 3
   • f(g(x)) = f(x - 3) = (x - 3)² = x² - 6x + 9

2. If f(x) = sin(x) and g(x) = cos(x), find (g ∘ f)(x):

   • f(x) = sin(x)
   • g(f(x)) = g(sin(x)) = cos(sin(x))

Step 4: Properties of Function Composition

• Function composition is not always commutative, meaning (f ∘ g)(x) is not necessarily equal to (g ∘ f)(x).

Example

• With f(x) and g(x) as defined above, (f ∘ g)(x) will yield a different result than (g ∘ f)(x).

Common Pitfalls

• Misunderstanding the order of operations. Remember to always apply the inner function first.
• Confusing the output of one function as the input for another without proper substitution.

Conclusion

In this tutorial, you have learned the basics of function composition, how to correctly compose two functions, and the significance of order in function composition. Practice with various functions will deepen your understanding and ability to solve complex mathematical problems. As a next step, try composing functions with different types and explore their properties to enhance your skills further.