Menentukan Salah Satu Fungsi Jika Diketahui Fungsi Komposisinya

Introduction

This tutorial focuses on determining one function when the composition of functions is known. Understanding function composition is essential in algebra, and this guide will break down the process into manageable steps, providing examples and practical tips to enhance your understanding.

Step 1: Understand Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. It is denoted as (f ∘ g)(x), which means f(g(x)).

Key Points

• Notation: f(g(x)) represents the composition of functions f and g.
• Order Matters: The result of f(g(x)) is generally not the same as g(f(x)).

Step 2: Identify the Given Functions

In most problems, you will be given the composition of functions and one of the functions. Your task is to find the other function.

Example Given

Suppose you know:

• (f ∘ g)(x) = x^2 + 1
• g(x) = x + 2

Practical Tip

Always write down the functions you know to visualize the problem better.

Step 3: Substitute and Simplify

To find the unknown function, substitute the known function into the composition.

Steps to Follow

1. Substitute g(x) into the composition:
   • (f ∘ g)(x) = f(g(x)) = f(x + 2)
2. Set the composition equal to the given expression:
   • f(x + 2) = x^2 + 1

Step 4: Solve for the Unknown Function

Now, you need to express f in terms of x.

Rearranging the Equation

1. Let y = x + 2, thus x = y - 2.
2. Replace x in the expression:
   • f(y) = (y - 2)^2 + 1

Expand and Simplify

• f(y) = y^2 - 4y + 4 + 1
• f(y) = y^2 - 4y + 5

Step 5: Replace with Original Variable

Finally, replace y back with x to express f in its original variable:

• f(x) = x^2 - 4x + 5

Conclusion

By following these steps, you can determine one function from its composition with another. The key is to clearly identify known functions, substitute, and rearrange the equations accordingly. Practice with various examples to strengthen your understanding of function composition and how to manipulate expressions effectively.

Next Steps

• Try more complex examples involving different types of functions.
• Explore how to determine g(x) if f(x) is known along with the composition.
• Review the properties of function composition for deeper insights.