Real functions | Part-1

Function as a Special Type of Relation

A function is a specific kind of relation in mathematics where every element in the domain is associated with exactly one element in the range. While all functions are relations, not all relations are functions.

Definition of a Function

A function f from a set A (domain) to a set B (codomain) is a rule that assigns each element in A to a unique element in B.

Notation: f: A → B

Condition: No element in the domain has more than one image in the codomain.

Relation and Ordered Pair

Functions can be represented using ordered pairs (x, y), where x belongs to the domain and y belongs to the range. Each value of x in a function is linked to a single, unique value of y.

Example: f = {(1, 2), (2, 4), (3, 6)} is a function because each input has one unique output.

Mapping Representation

Mapping diagrams are commonly used to represent functions by linking every element in the domain to one specific element in the range.

  Domain     →     Range
    1        →       2
    2        →       4
    3        →       6
  

Every item in the domain (left side) is paired with a single, unique item in the range (right side).

Illustrations of Functions

• Example 1: Let f(x) = x + 5. Given the domain {1, 2, 3}, the function evaluates as follows:
  • f(1) = 6
  • f(2) = 7
  • f(3) = 8
• Example 2: Let f(x) = x² for domain {–2, –1, 0, 1, 2}
  • f(–2) = 4
  • f(–1) = 1
  • f(0) = 0
  • f(1) = 1
  • f(2) = 4

Key Points

• Every function is a relation, but not every relation is a function.
• A function assigns only one output to each input.
• Functions can be represented in many forms: ordered pairs, mapping diagrams, or algebraic rules.