MMW CHAPTER 5

5.2.1 Definition and Properties Definition 5.2.1: A function is explicitly defined as a relation that assigns each input x in set A to exactly one output y in set B. In this context, it is vital to note that no two ordered pairs within the function can share the same first element, preventing ambiguity in the mapping.

Example 5.2.2: Functions such as F1 = {(1,2),(3,4),(5,6)} clearly illustrate unique mappings where each input value corresponds to a distinct output value.

Arrow Diagrams: These visual tools support the demonstration of valid functions, showcasing how inputs distinctly relate to outputs.

Vertical Line Test: This test serves as a graphical verification method; if any vertical line intersects a graph at more than one point, the relation fails to qualify as a function.

Elementary Functions:

  • Polynomial Function: P(x) = a_nx^n + ... + a_1x + a_0, where each coefficient determines the weight of the respective x term.

  • Constant Function: P(x) = a_0; characterized by a horizontal line graph, indicating a fixed output for all inputs.

  • Zero Function: P(x) = 0; represents a function where all outputs are constant at zero.

  • Linear Function: P(x) = a_1*x + a_0; results in a straight line graph, demonstrating a consistent rate of change.

  • Quadratic Function: P(x) = a_2x² + a_1x + a_0; produces a parabolic graph that can open upwards or downwards based on the sign of a_2.

  • Cubic Function: P(x) = a_3x³ + a_2x² + a_1*x + a_0; produces curves and has points of inflection, allowing for more complex behavior than quadratic functions.

  • Rational Function: P(x) = (p(x))/(q(x)); where both p(x) and q(x) are polynomial functions, and q(x) ≠ 0. These functions may exhibit asymptotic behavior.

  • Root Function: P(x) = √(x); represents the principal square root of x, producing half of a sideways parabola.

  • Piecewise Function: A function defined by multiple sub-functions, each applicable to a specified interval of the function's domain. Example: [ f(x) = \left{ \begin{array}{ll} x^2 & \text{if } x < 0 \ x + 1 & \text{if } x \geq 0 \end{array} \right. ]

  • Absolute Value Function: P(x) = |x|; gives the non-negative value of x, producing a V-shaped graph symmetrical about the y-axis.