2.1 Graphs of Basic Functions and Relations; Symmetry

Continuity of Functions

  • Definition of Continuity: A function is continuous over an interval of its domain if its hand-drawn graph can be sketched without lifting the pencil from the paper.

Discontinuous Functions

  • Not all functions are continuous.

  • Examples of Discontinuity:

    • Hole in the Graph: A point in the graph where the function is not defined (point of discontinuity).

    • Vertical Asymptote: A point where the function approaches infinity or becomes undefined.

  • In both cases, the graph cannot be drawn continuously without lifting the pencil.

Analyzing Continuous Functions

  • Graph Types:

    • Graph 1: Parabolic curve (quadratic function) - Continuous over all real numbers (entire domain).

    • Graph 2: Discontinuous at x = 3 - Continuous intervals: (-∞, 3) and (3, ∞).

    • Graph 3: Vertical asymptote at x = -2 - Continuous intervals: (-∞, -2) and (-2, ∞).

Increasing, Decreasing, and Constant Functions

  • Function Behavior:

    • A continuous function can be increasing, decreasing, or constant over its interval.

    • Definitions:

      • Increasing: Function rises from left to right (positive slope).

      • Decreasing: Function falls from left to right (negative slope).

      • Constant: Output remains the same throughout the interval (horizontal line).

  • Open Interval Notation: Endpoints are not included (denoted by parentheses).

Formal Definitions of Function Behavior

  • For an open interval I:

    • Increasing: If x₁ < x₂, then f(x₁) < f(x₂).

    • Decreasing: If x₁ < x₂, then f(x₁) > f(x₂).

    • Constant: For all x₁, x₂ in I, f(x₁) = f(x₂.

Basic Functions and Their Graphs

  • Learning Parent Functions:

    • Identity Function:

      • Linear function: slope = 1, y-intercept = 0.

      • Domain: all real numbers.

      • Increasing over entire domain, continuous.

    • Squaring Function:

      • Graph is a parabola; Domain: all reals, Range: [0, ∞).

      • Decreases from (-∞, 0) and increases from (0, ∞).

      • Symmetric with respect to the y-axis (even function).

    • Cubing Function:

      • Continuous over all reals, increasing everywhere.

      • Symmetric with respect to the origin (odd function).

    • Square Root Function:

      • Domain: [0, ∞), increasing and continuous.

    • Absolute Value Function:

      • Domain: all reals, Range: [0, ∞).

      • Symmetric with respect to the y-axis.

Symmetry of Functions

  • Symmetry Definitions:

    • With Respect to Y-Axis: If f(-x) = f(x), symmetric about the y-axis (even function).

    • With Respect to Origin: If f(-x) = -f(x), symmetric about the origin (odd function).

Even and Odd Functions

  • Even Function: f(-x) = f(x), symmetric about the y-axis.

  • Odd Function: f(-x) = -f(x), symmetric about the origin.

  • Functions can be even, odd, or neither.

Examples of Even, Odd, or Neither

  1. Quartic Function: Proved even, symmetric about y-axis.

  2. Cubic Function: Proved odd, symmetric about the origin.

  3. Quadratic Function: Neither even nor odd, not symmetric about either axis.

Summary of Key Concepts

  • Understanding continuity, increasing/decreasing functions, symmetry, and the properties of basic functions is fundamental to graphing and analyzing functions.

  • Recognizing types of symmetry aids in determining whether functions are even, odd, or neither.