Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 8 | - 3$, where $x - 8$
A) Domain: $[ - 8 , \infty )$; Range: $[ - 3 , \infty )$
B) Domain: $( - \infty , - 3 ]$; Range: $[ - 8 , \infty )$
C) Domain: $[ - 8 , - 3 ]$; Range: $[ - 8 , \infty )$
D) Domain: $( - \infty , - 8 ]$; Range: $[ 3 , \infty )$
E) Domain: $( - \infty , \infty ) ;$ Range: $( - \infty , \infty )$

Question

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ $f ( x ) = - | x + 8 | - 3$, where $x > - 8$
A) Domain: $[ - 8 , \infty )$; Range: $[ - 3 , \infty )$
B) Domain: $( - \infty , - 3 ]$; Range: $[ - 8 , \infty )$
C) Domain: $[ - 8 , - 3 ]$; Range: $[ - 8 , \infty )$
D) Domain: $( - \infty , - 8 ]$; Range: $[ 3 , \infty )$
E) Domain: $( - \infty , \infty ) ;$ Range: $( - \infty , \infty )$

Quizplus · Accepted Answer

To find the domain of the inverse function, we need to look at the range of the original function. Since the range of $f(x)$ is $[-\infty, -3]$, the domain of $f^{-1}(x)$ is $(-\infty, -3]$ (the inverse function will only output values that are in the range of $f(x)$).

To find the range of the inverse function, we need to look at the domain of the original function. The given condition $x>-8$ means that the graph of the function is shifted to the left by 8 units, but it does not affect the range. The absolute value and the negative sign mean that the maximum value of the function is -3, so the range of $f(x)$ is $[-\infty, -3]$. Therefore, the range of $f^{-1}(x)$ is $[-8,\infty)$.

The correct answer is B, since it gives the correct domain and range.

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

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