The function f(x) = x² - 2 is one-to-one on the domain (x $\le$ 0) .Find f ^-1(x). A) $f ^ { - 1 } ( x ) = - \sqrt { x + 2 }$ B) $f ^ { - 1 } ( x ) = \frac { 1 } { x ^ { 2 } - 2 }$ C) $f ^ { - 1 } ( x ) = \sqrt { x + 2 }$ D) $f ^ { - 1 } ( x ) = \sqrt { x - 2 }$ E)f ^-1(x) = x² + 2

Question

Quizplus · Accepted Answer

Given $f(x) = x^2 - 2$, to find $f^{-1}(x)$, we swap $x$ and $y$ and solve for $y$. So, $x = y^2 - 2$, which leads to $y^2 = x + 2$. Since the domain is $x \leq 0$, the inverse function must produce negative outputs (or non-positive when considering 0), leading to $f^{-1}(x) = -\sqrt{x + 2}$.

The Function F(x) = X2 - 2 Is One-To-One on the Domain

The Function F(x) = X² - 2 Is One-To-One on the Domain