Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$.
$f ( x ) = - | x + 6 | + 2$, where $x - 6$
A) Domain: $[ - 6 , \infty )$; Range: $[ 2 , \infty )$
B) Domain: $( - \infty , 2 ]$; Range: $[ - 6 , \infty )$
C) Domain: $[ - 6,2 ]$; Range: $[ - 6 , \infty )$
D) Domain: $( - \infty , - 6 ]$; Range: $[ - 2 , \infty )$
E) Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$

Question

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$.
$f ( x ) = - | x + 6 | + 2$, where $x > - 6$
A) Domain: $[ - 6 , \infty )$; Range: $[ 2 , \infty )$
B) Domain: $( - \infty , 2 ]$; Range: $[ - 6 , \infty )$
C) Domain: $[ - 6,2 ]$; Range: $[ - 6 , \infty )$
D) Domain: $( - \infty , - 6 ]$; Range: $[ - 2 , \infty )$
E) Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$

Quizplus · Accepted Answer

The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$. For $f(x) = -|x+6| + 2$, the output decreases as $x$ increases, starting from $2$ when $x > -6$, thus the range of $f$ is $(-\infty, 2]$, making it the domain of $f^{-1}$. The original domain of $f$ is $(-6, \infty)$, which becomes the range of $f^{-1}$.

Determine the Domain and Range of the Inverse Function f−1f ^ { - 1 }f−1

Determine the Domain and Range of the Inverse Function $f ^ { - 1 }$