Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 7 | - 1$, where $x - 7$
A) Domain: $[ - 7 , \infty )$; Range: $[ - 1 , \infty )$
B) Domain: $( - \infty , - 1 ]$; Range: $[ - 7 , \infty )$
C) Domain: $[ - 7 , - 1 ]$; Range: $[ - 7 , \infty )$
D) Domain: $( - \infty , - 7 ]$; Range: $[ 1 , \infty )$
E) Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$

Question

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 7 | - 1$, where $x > - 7$
A) Domain: $[ - 7 , \infty )$; Range: $[ - 1 , \infty )$
B) Domain: $( - \infty , - 1 ]$; Range: $[ - 7 , \infty )$
C) Domain: $[ - 7 , - 1 ]$; Range: $[ - 7 , \infty )$
D) Domain: $( - \infty , - 7 ]$; Range: $[ 1 , \infty )$
E) Domain: $( - \infty , \infty )$; Range: $( - \infty , \infty )$

Quizplus · Accepted Answer

To find the inverse, we first solve for x in terms of y:
\begin{align*}
y & = -|x+7|-1 \
y+1 & = -|x+7| \
\frac{y+1}{-1} & = |x+7| \
|x+7| & = -y-1 \
\end{align*}
Since a absolute value can never be negative, we know that $-y-1 \geq 0$. Solving for y, we get $y \leq -1$. Thus, the range of $f^{-1}$ is $[-7,\infty)$ and the domain of $f^{-1}$ is $(-\infty,-1]$, since $x > -7$ in the original function. The only answer choice that matches these conditions is B.

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

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