Understanding Functions: Definitions and Properties

Sep 1, 2024

Functions and Their Properties

Definition of a Function

  • A function is a relation where each element of the domain maps to exactly one element of the range.
    • Every x has only one y.
    • Example: If x1 maps to y1, x2 maps to y2, that forms a function.
  • A mapping diagram can help visualize if a relation is a function.
    • Example of a non-function: If x1 maps to both y1 and y3, it is not a function.

Determining Functions

  • Mapping Diagram: Useful for visualizing whether each x maps to only one y.
  • Vertical Line Test:
    • A visual test where a vertical line swept across the graph should not intersect it more than once at any location to confirm it's a function.

Domain and Range

  • Domain: Set of all possible input values (x-values).
  • Range: Set of all possible output values (y-values).

Domain and Range of Algebraic Functions

  • Square Roots:
    • Cannot have a negative number under the square root in real numbers.
    • Example: For sqrt(x + 4), x + 4 >= 0 implies x >= -4.
  • Fractions:
    • Cannot divide by zero.
    • Example: For 1/(x - 3), x cannot be 3.
  • Interval Notation:
    • Domain: x ≥ -4 is represented as [-4, ∞).
    • Excluding 3: [0, 3) ∪ (3, ∞).
  • Range:
    • Use graphing calculator for visualization.
    • Example: Exclude y = 0 range for functions crossing x-axis.

Contextual Domain and Range

  • Problem-specific restrictions:
    • Volume of a Sphere:
      • Domain: Radius r cannot be zero or negative.
      • Range: Volume cannot be negative.
    • Domain and Range for such problems are often deduced using context and common sense.
    • Example: Domain and Range for volume of a sphere are both (0, ∞).

Key Points

  • A function is defined by its unique x-y mapping.
  • Domain and range are crucial for understanding the behavior of functions.
  • Use graphical methods and contextual insights to determine domain and range.