Maths » Collège » Calcul numérique
Comment calculer un antécédent quand on connait l'image par une fonction affine.
Comment calculer l'antécédent d'une image par une fonction affine ?
Le problème.
On connaît l'image d'un nombre par une fonction affine et on désire savoir quel antécédent a "renvoyé" cette image.
Les propriétés.
La seule propriété à connaître est la définition d'une fonction affine et la notation employée pour écrire l'image d'un nombre :
Soit a et b deux nombres donnés, la fonction qui, à tout nombre x, associe le nombre ax + b s'appelle fonction affine, on la note 
On peut aussi définir une fonction en donnant le mode de calcul de l'image à partir de l'antécédent x : 
où 
est l'image de x par f.
La méthode.
Comme l'image et la fonction affine sont données, il suffit de remplacer 
par sa valeur et de résoudre l'équation.
L'exemple.
On sait que l'image par la fonction affine f définie par 
est – 2 . On veut connaître l'antécédent de – 2 par cette fonction.
On a 
et 
donc
Pour connaître l'antécédent x il suffit de résoudre cette équation :
L'antécédent recherché est 
On vérifie :
C'est exact !
Last update: 2006-02-10 10:49
Author: Henry Serge
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