Functions and Relations: Domain and Range

Relations

• A relation is a set of ordered pairs (x, y).
• Each ordered pair consists of an input (x) and an output (y).

Functions

• A function is a special type of relation where each input (x) corresponds to exactly one output (y).
• Notation: f(x) denotes the function value for input x.

Domain

• The domain of a relation or function is the complete set of possible input values (x-values).
• To find the domain:
  • Identify all potential x-values from the set of ordered pairs.
  • Consider restrictions (e.g., division by zero, square roots of negative numbers).

Range

• The range of a relation or function is the complete set of possible output values (y-values).
• To find the range:
  • Identify all potential y-values from the set of ordered pairs.
  • Look for any restrictions on output values.

Key Concepts

• Example of Domain and Range:
  • For the relation {(1, 2), (3, 4), (5, 6)}, the domain is {1, 3, 5} and the range is {2, 4, 6}.
• Vertical Line Test:
  • A visual method to determine if a relation is a function. If a vertical line intersects the graph of the relation at more than one point, it is not a function.

Relations

• A relation consists of ordered pairs (x, y), associating inputs (x) with outputs (y).

Functions

• A function is a specific type of relation where each input (x) links to one and only one output (y).
• Function notation is represented as f(x), indicating the function's output value for a given input x.

Domain

• The domain refers to the complete set of all possible input values (x-values) for a relation or function.
• To determine the domain:
  • Identify all x-values from the ordered pairs.
  • Take into account any restrictions, such as avoiding division by zero or square roots of negative numbers.

Range

• The range represents the complete set of possible output values (y-values) for a relation or function.
• To determine the range:
  • Identify all y-values from the ordered pairs.
  • Consider any restrictions affecting output values.

Key Concepts

• Example of Domain and Range:

  • For the relation {(1, 2), (3, 4), (5, 6)}, the identified domain is {1, 3, 5} and the range is {2, 4, 6}.

• Vertical Line Test:

  • A technique for determining if a relation qualifies as a function.
  • If a vertical line intersects a graph of the relation at more than one point, it indicates that the relation is not a function.