The function $f ( x ) = x ^ { 2 } - 5$ is one-to-one on the domain $$( x \leq 0 )$$ .Find $f ^ { - 1 } ( x )$ . ?
A) $f ^ { - 1 } ( x ) = x ^ { 2 } + 5$
B) $f ^ { - 1 } ( x ) = \sqrt { x + 5 }$
C) $f ^ { - 1 } ( x ) = - \sqrt { x + 5 }$
D) $f ^ { - 1 } ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$
E) $f ^ { - 1 } ( x ) = \sqrt { x - 5 }$

Question

Quizplus · Accepted Answer

To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ in the original function $y = x^2 - 5$ and solve for $x$. This gives us $x = y^2 - 5$, which when solving for $y$, we get $y = \sqrt{x + 5}$. However, since the domain is $x \leq 0$, the inverse function must produce negative outputs for $x$, leading to $f^{-1}(x) = -\sqrt{x + 5}$.

The Function f(x)=x2−5f ( x ) = x ^ { 2 } - 5f(x)=x2−5

The Function $f ( x ) = x ^ { 2 } - 5$