Mapping of Functions

1 Understanding Function Mapping

• Definition: A function is a relation where each input (from the domain) is mapped to exactly one output (in the range).
• Mapping Representation: Functions can be represented using:
  • Arrow Diagrams
  • Ordered Pairs
  • Tables of Values
  • Graphical Representation

2 Types of Mappings

• One-to-One (Injective) Mapping

  • Each input has a unique output.
  • Example: f(x)=x+2 (No two inputs map to the same output).
  • Graph: Passes the horizontal line test.

• Many-to-One Mapping

  • Multiple inputs map to the same output.
  • Example: f(x)=x^2 (Both x = 2 and x=−2 give the same output).

• Onto (Surjective) Mapping

  • Every element in the range has at least one preimage in the domain.
  • Example: f(x)=2x (Every real number has a corresponding x).

• One-to-One and Onto (Bijective) Mapping

  • A function that is both one-to-one and onto.
  • Example: f(x)=x+3f(x) = x + 3 (Each input has a unique output, and every output is covered).

3 Function Notation & Representation

• Function Notation: f(x)f(x) represents a function where xx is the input.
• Example:
  • f(x)=2x+1f(x) = 2x + 1
  • Input: x=3x = 3, Output: f(3)=2(3)+1=7f(3) = 2(3) + 1 = 7.
• Set Representation:
  • Domain: Set of all possible input values.
  • Range: Set of all possible output values.

4 Vertical Line Test

• Purpose: To check if a given graph represents a function.
• Test: If a vertical line crosses the graph at more than one point, the relation is not a function.

5 Real-Life Applications of Function Mapping

• Economics: Demand and supply functions.
• Physics: Velocity-time functions.
• Engineering: Input-output systems.
• Computer Science: Algorithmic mappings (Hash functions).