Обратна функция

Функцията $f$ и обратната ѝ $f^{-1}$ . Ако $f(a)=3$ , то $f^{-1}(3)=a$ .

Обратна функция $g(y)$ на функцията $f(x)$ се нарича такава функция, за която $g(f(x))=x$ за всяко x. Най-често за обратна функция се използва означението $f^{-1}(y)$ . Функцията $f(x)$ има обратна функция тогава и само тогава, когато тя е биекция. ^[1]^[2]

Дефиниция на обратна функция

Ако функцията $y=f(x)$ е дефинирана и обратима в множеството $D$ и приема стойности в множеството $M$ , тогава функцията $x=f^{-1}(y)$ се нарича обратна на $y=f(x)$ , ако за всяко $y$ , принадлежащо на $M$ , съществува единствено $x$ , принадлежащо на $D$ , за което $y=f(x)$ .

Обратима функция

Казваме, че функцията $y=f(x)$ е обратима в множеството $D$ , ако за всеки 2 стойности $x1$ , $x2$ , принадлежащи на $D$ , от това, че $x1$ е различно от $x2$ следва, че $f(x1)$ е различно от $f(x2)$ .

Всяка монотонна функция е обратима, защото от монотонността следва, че функцията е биекция.

Свойство 1

Ако функцията $y=f(x)$ е монотонно растяща/намаляваща в множеството $D$ , то обратната и функция $x=f^{-1}(y)$ е монотонно растяща/намаляваща в $M$ .

Свойство 2

Графиките на дадена функция и нейната обратна са симетрични спрямо ъглополовящите на първи и трети квадрант.

Примери

Обратната функция на показателната функция е логаритмичната функция.

Тригонометрични функции
Функция	Обратна функция
$sin(x)$	$arcsin(x)$
$cos(x)$	$arccos(x)$
$tg(x)$	$arctg(x)$
$cotg(x)$	$arccotg(x)$
$sec(x)$	$arcsec(x)$
$cosec(x)$	$arccosec(x)$

Примерна задача

Да се намери обратната функция на функцията y = $x^{2}$ , x ≥ 0.

Решение:

Функцията y = $x^{2}$ , x ≥ 0 е строго растяща, защото от 0 ≤ x1 < x2 следва f(x1)-f(x2) = $x1^{2}$ - $x2^{2}$ = (x1 – x2)(x1 + x2) < 0, т.е. f(x1) < f(x2). Това означава, че функцията y = $x^{2}$ при x ≥ 0 е обратима. От y = $x^{2}$ , x ≥ 0 намираме x = √y. Изразът √y има смисъл, защото y ≥ 0. Следователно обратната функция на функцията Функцията y = $x^{2}$ за x ≥ 0 е x = √y, y ≥ 0.

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