Symmetric Functions with Respect to a Point (a,b) and Its Properties that Generalized from Properties of Odd Functions

Abstract

The function f: R \to R is said to be an odd function if  f (-x) = -f(x) for every x \in R. The graph of an odd function is symmetry with respect to origin, or point (0,0) . The propose of this study is to observe some properties of symmetrical functions which are generalize from some properties of odd functions. Some of the results obtained include a linear combination of two functions symmetrical with respect to the point  (a,b) is a functions of symmetrical with respect to the point (a,2b). An integral functions of symmetrical with respect to the point (a,b) on a closed interval [a-c,a+c] is 2bc for any real number c. Moreover product of scalars with functions of symmetrical with respect to the point (a,b)  is a functions of symmetrical with respect to the point (a,\alpha b)  for every \alpha real numbers. Furthermore the addition n-symmetrical of functions with respect to the point (a,b) is a series of functions of symmetrical with respect to the point (a,nb).

he function  is said to be an odd function if  for every . The graph of an odd function is symmetry with respect to origin, or point . The propose of this study is to observe some properties of symmetrical functions which are generalize from some properties of odd functions. Some of the results obtained include a linear combination of two functions symmetrical with respect to the point  is a functions of symmetrical with respect to the point . An integral functions of symmetrical with respect to the point  on a closed interval  is  for any real number . Moreover product of scalars with functions of symmetrical with respect to the point  is a functions of symmetrical with respect to the point  for every  real numbers. Furthermore the addition -symmetrical of functions with respect to the point  is a series of functions of symmetrical with respect to the point .