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

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) $f ^ { - 1 } ( x ) = \sqrt { x - 5 }$

Quizplus · Accepted Answer

To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ in the original function and solve for $y$. Starting with $y = x^2 - 5$, we swap to get $x = y^2 - 5$, and then solve for $y$, giving $y = \sqrt{x + 5}$. However, since the function is defined as one-to-one on the domain $x \leq 0$, the correct inverse must produce negative outputs for the given domain, making the correct answer $y = -\sqrt{x + 5}$.

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

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