Real Valued Functions - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

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Chapter 9

Efficiency of Algorithms

Real Valued Functions

Example

Power Functions

• Definition
  • Let a be any nonnegative number. Define pa , the power function with exponent a, as follows: pa (x) = xa^ for each nonnegative real number x.

Graphing Function on Integers

• A real-valued function may be graphed on a

set of integers.

Multiple of a Function

• Definition
  • Let f be a real-valued function of a real variable and let M be any real number. The function Mf, called the multiple of f by M, is the real-valued function with the same domain as f that is defined by the rule

(Mf)(x) = M* ((f(x)) for all x in the domain of f

Increasing & Decreasing Functions

f ( x ) = x =

x , if x ≥ 0

− x , if x < 0

Increasing & Decreasing Function

• Definition
  • Let f be a real-valued function defined on a set of real numbers, and suppose the domain of f contains a set S. We say that f is increasing on the set S if, and only if, for x 1 and x 2 in S, if x 1 < x 2 then f(x 1 ) < f(x 2 ).

We say that f is decreasing on the set S if, and only if, for x 1 and x 2 in S, if x 1 < x 2 then f(x 1 ) > f(x 2 ).