The Fuji X-T1 Review. Fuji creates the Best X to date! – Steve Huff PhotoEven and Odd Functions
Even and Odd Functions
They are special types of functions
Even Functions
A function is &even& when:
f(x) = f(&x) for all x
In other words there is
(like a reflection):
This is the curve f(x) = x2+1
They got called &even& functions because the functions x2, x4, x6, x8, etc behave like that, but there are other
functions that behave like that too, such as cos(x):
Cosine function: f(x) = cos(x)
It is an even function
exponent does not always make an
even function, for example (x+1)2 is not an even function.
Odd Functions
A function is &odd& when:
&f(x) = f(&x) for all x
Note the minus in front of f(x): &f(x).
And we get
This is the curve f(x) = x3&x
They got called &odd& because the functions x, x3, x5, x7, etc behave like that, but there are other
functions that behave like that, too, such as sin(x):
Sine function: f(x) = sin(x)
It is an odd function
exponent does not always make an
odd function, for example x3+1 is not an odd function.
Neither Odd nor Even
Don't be misled by the names &odd& and &even& ... they are just names ... and a function does not have to be even or odd.
In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:
This is the curve f(x) = x3&x+1
It is not an odd function, and it is not an
even function either.
It is neither odd nor even
Even or Odd?
Example: is f(x) = x/(x2&1) Even or Odd or neither?
Let's see what happens when we substitute &x:
f(&x) = (&x)/((&x)2&1)
=&x/(x2&1)
So f(&x) = &f(x) , which makes it
an Odd Function
Even and Odd
The only function that is even&and odd is f(x) = 0
Special Properties
The sum of two even functions is even
The sum of two odd functions is odd
The sum of an even and odd function is neither even nor odd (unless one function is zero).
Multiplying:
The product of two even functions is an even function.
The product of two odd functions is an even function.
The product of an even function and an odd function is an odd function.
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a r t i a l m e t r i
e m e t r i
c 1 2 3 Alain Guénoche
, Bruno Leclerc
, Vladimir Makarenkov 1Institut de Mathématiques de Luminy, 163 avenue de Luminy, F-13009 MARSEILLE, FRANCE, guenoche @iml.univ-mrs.fr 2Centre d'Analyse et de Mathématique Sociales, école des Hautes études en Sciences Sociales, 54 bd Raspail, F-75270 PARIS CEDEX 06, FRANCE, leclerc @ehess.fr 3Département de Sciences Biologiques, Université de Montréal, C.P. 6 128, succ. Centre-ville, Montréal, Québec H3C 3J7, CANADA, and Institute of Control Sciences, 65 Profsoyuznaya, Moscow 117806, RUSSIA, makarenv @magellan.umontreal.ca Revised, April 2002 Abstract.
completion
to additive
and proved it to be NP-complete: given a partial dissimilarity d on a finite
set X , does there exist a tree metric extending d to all pairs of elements of X . We use a previously described simple
phylogenetic
reconstruction,
to partial
dissimilarities,
to characterize some classes of polynomial instances of MCA and of a related problem. We point out that these problems admit many other polynomial instances. We focus particularly on two classes of
generalized cycles, together
corresponding maximal
2-trees and
2d- trees . Résumé. Farach, Kannan et Warnow
ont posé le problème MCA
matrix completion to additive
suivant et ont démontré sa NP-complétude : étant donné une dissimilarité d partielle sur un ensemble fini X , est-il possible de l'étendre en une distance d'arbre définie sur toutes les paires d'éléments
reconstruction
phylogénétique, précédemment décrite, et
son extension aux dissimilarités partielles pour caractériser des classes d'instances polynomiales de MCA
d'un problème voisin. Nous montrons qu'en fait
beaucoup d
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