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

To find the inverse function, we need to solve for x in terms of y:
$$y = -|x+6|+2$$
$$y-2=-|x+6|$$
$$|x+6|=2-y$$
Since x > -6, x + 6 > 0, so we can remove the absolute value and get:
$$x+6=y-2 \Rightarrow x=y-8$$
Thus, the inverse function is:
$$f^{-1}(y)=y-8$$
The domain of the inverse function is the range of the original function, which is $[ - 6 , \infty )$ since $x>-6$. The range of the inverse function is the domain of the original function, which is $( - \infty , 2 ]$ since the maximum value of the original function is 2. Therefore, the answer is choice B.

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

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