FUNCTIONS - DISCRETE MATHEMATICS

Introduction

This tutorial provides a comprehensive guide on functions in discrete mathematics, covering how to write functions, understand their terminology, and compose them. Whether you are a student or someone interested in mathematics, this guide will help you grasp the basics of functions and their applications.

Step 1: Understanding Functions

• Definition: A function is a relation that assigns exactly one output for each input from a set called the domain.
• Notation: Functions are usually denoted by letters such as f, g, or h.
• Example:
  • If f(x) = x + 2, then for an input of 3, the output is f(3) = 3 + 2 = 5.

Key Terminology

• Domain: The set of all possible input values for a function.
• Codomain: The set of potential output values.
• Range: The actual set of output values produced by the function.

Step 2: Writing Functions

• Function Definition: To define a function, specify the input-output relationship clearly.
• Example:
  • Let f(x) = x^2. This means for any input x, the output is the square of x.
• Practical Advice: Always clarify the domain when defining a function. For instance, if f(x) = 1/x, the domain cannot include 0.

Step 3: Composing Functions

• Definition of Composition: Composing functions involves using the output of one function as the input for another.
• Notation: The composition of functions f and g is denoted as (f ∘ g)(x) = f(g(x)).
• Example:
  • Let f(x) = x + 1 and g(x) = 2x. Then, (f ∘ g)(x) = f(g(x)) = f(2x) = 2x + 1.

Practical Tips

• Ensure the output of the inner function matches the domain of the outer function.
• Check your work by substituting values to confirm the outputs align.

Step 4: Visualizing Functions

• Graphing: Visual representation helps understand functions better.
• Tools: Use graphing calculators or software like Desmos to visualize functions and their compositions.
• Common Pitfalls: Misreading graphs can lead to incorrect interpretations of a function's behavior.

Conclusion

By understanding the definitions, writing functions correctly, composing them, and visualizing their behavior, you will have a solid grasp of functions in discrete mathematics. For further learning, explore additional resources or practice with different types of functions and their compositions. Engage with online platforms or textbooks to deepen your understanding of this fundamental concept in mathematics.