Introduction to Functions and Relations

Overview

In this section, we’ll explore the basic concepts of functions and relations, understanding how they help us represent relationships between two quantities. A relation is a set of ordered pairs, while a function is a specific type of relation where each input has exactly one output.

Key Concepts

  • Relation: A set of ordered pairs (x, y) that associates elements from one set with elements of another set.
  • Function: A type of relation where each input (or x-value) is related to exactly one output (or y-value).
  • Domain: The set of all possible input values (x-values) for a relation or function.
  • Range: The set of all possible output values (y-values) for a relation or function.
  • Function Notation: Functions are commonly written in the form \( f(x) \), where \( f \) represents the function and \( x \) is the input variable.

Examples

Consider the relation: \( \{(1, 3), (2, 5), (3, 7)\} \). Each input has one output, so this is a function. In function notation, we could represent this as \( f(1) = 3 \), \( f(2) = 5 \), and \( f(3) = 7 \).

Practice Questions

  1. Determine whether the following set of ordered pairs is a function: \( \{(2, 4), (3, 6), (2, 8), (5, 10)\} \).
    Solution

    No, this is not a function because the input \( x = 2 \) has two different outputs (4 and 8).

  2. Identify the domain and range of the relation \( \{(0, 1), (1, 3), (2, 5), (3, 7)\} \).
    Solution

    Domain: \( \{0, 1, 2, 3\} \)
    Range: \( \{1, 3, 5, 7\} \)

  3. For the function \( f(x) = x^2 + 1 \), find \( f(2) \).
    Solution

    Plugging in \( x = 2 \), we get \( f(2) = 2^2 + 1 = 5 \).

  4. Which of the following mappings represents a function?
    A) \( \{(1, 2), (2, 4), (3, 6), (3, 7)\} \)
    B) \( \{(1, 5), (2, 6), (3, 7), (4, 8)\} \)
    Solution

    Mapping B represents a function because each input has exactly one output, while mapping A does not since input \( x = 3 \) has two different outputs (6 and 7).

  5. Express the relation \( y = 2x + 3 \) in function notation.
    Solution

    In function notation, this relation can be written as \( f(x) = 2x + 3 \).

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