Composition of Functions | General Mathematics | Grade 11

Introduction

This tutorial will guide you through the concept of composition of functions, a key topic in General Mathematics for Grade 11 students. Understanding how to compose functions is essential for solving complex mathematical problems and lays the foundation for more advanced topics in mathematics.

Step 1: Understanding Functions

• A function is a relation that assigns exactly one output for each input.
• Common notation for a function is f(x), where:
  • f represents the function name.
  • x is the input variable.
• Example: If f(x) = 2x + 3, then for an input of 2, f(2) = 2(2) + 3 = 7.

Step 2: What is Function Composition

• Function composition involves combining two functions to form a new function.
• Notation for composition is (f ∘ g)(x), which means you apply g first and then f.
• This can also be expressed as f(g(x)).

Step 3: Steps to Compose Functions

1. Identify the two functions you want to compose. For example:
   • Let f(x) = 2x + 3
   • Let g(x) = x^2
2. Determine the order of composition. Decide which function will be applied first:
   • For (f ∘ g)(x), apply g first, then f.
3. Substitute g into f:
   • Write f(g(x)): f(g(x)) = f(x^2)
   • Replace x in f with x^2: f(x^2) = 2(x^2) + 3 = 2x^2 + 3.

Step 4: Example of Composition

• Using the previously defined functions:
  • g(f(x)) would be calculated as follows:
    1. Substitute f into g: g(f(x)) = g(2x + 3).
    2. Replace x in g with f(x): g(2x + 3) = (2x + 3)^2.
    3. Expand: (2x + 3)(2x + 3) = 4x^2 + 12x + 9.

Step 5: Graphical Interpretation

• Understand that composing functions can also be visualized graphically.
• The graph of the composite function (f ∘ g)(x) represents the output of applying g and then f.
• Use graphing software or a graphing calculator to visualize the functions and their composition.

Step 6: Practice Problems

• To reinforce your understanding, try these practice problems:
  1. Let f(x) = x + 2 and g(x) = 3x. Find (f ∘ g)(x).
  2. Let f(x) = x^2 and g(x) = x - 1. Find (g ∘ f)(x).
• Solve these by following the composition steps outlined above.

Conclusion

In this tutorial, you learned about the composition of functions, how to compose them step-by-step, and the importance of understanding function relationships. Practice with various functions to strengthen your skills. As you progress, consider exploring more advanced topics like inverses and transformations of functions. For additional resources, check out related playlists linked in the video description.