COMPOSITION OF FUNCTIONS || GRADE 11 GENERALMATHEMATICS Q1

Introduction

This tutorial will guide you through the concept of composition of functions, an essential topic in Grade 11 General Mathematics. Understanding how to combine functions effectively is crucial for solving complex mathematical problems. This step-by-step guide will break down the process and provide practical examples to enhance your learning.

Step 1: Understand the Basics of Functions

• A function is a relation that assigns exactly one output for each input.
• Functions are usually written in the form f(x), where f is the function name and x is the input variable.
• Example: If f(x) = 2x + 3, then for x = 2, f(2) = 2(2) + 3 = 7.

Step 2: Learn About Composition of Functions

• Composition of functions involves combining two functions to create a new function.
• The notation for composition is (f ∘ g)(x) = f(g(x)).
• This means you first apply g to x, and then apply f to the result of g.

Step 3: Performing Composition of Functions

1. Identify the functions you want to compose. For example, let:
   • f(x) = 2x + 3
   • g(x) = x^2
2. Substitute g into f:
   • (f ∘ g)(x) = f(g(x)) = f(x^2)
3. Replace g(x) in the function f:
   • f(x^2) = 2(x^2) + 3
4. Simplify:
   • (f ∘ g)(x) = 2x^2 + 3

Step 4: Explore the Reverse Composition

• The reverse composition (g ∘ f)(x) is also important:

1. Substitute f into g:
   • (g ∘ f)(x) = g(f(x)) = g(2x + 3)
2. Replace f(x) in the function g:
   • g(2x + 3) = (2x + 3)^2
3. Simplify:
   • (g ∘ f)(x) = 4x^2 + 12x + 9

Step 5: Common Pitfalls to Avoid

• Ensure that you are applying the correct function in the correct order.
• Be cautious with parentheses; they can change the outcome.
• Always simplify your final expression for clarity.

Step 6: Real-World Applications

• Composition of functions can be applied in various fields like physics, economics, and engineering.
• For example, in physics, if one function represents distance as a function of time and another function represents speed as a function of distance, you can compose these functions to find speed as a function of time.

Conclusion

The composition of functions is a fundamental concept in mathematics that enables you to create new functions from existing ones. By practicing the steps outlined in this tutorial, you’ll gain a solid understanding of how to compose functions effectively. As a next step, try composing different functions on your own and explore their applications in real-world scenarios.