To determine the characteristics of the inverse of the given function, we need to understand the relationship between the original function and its inverse.
For a function [tex]\( f \)[/tex] with domain [tex]\( D \)[/tex] and range [tex]\( R \)[/tex]:
- The inverse function [tex]\( f^{-1} \)[/tex] will have domain [tex]\( R \)[/tex] and range [tex]\( D \)[/tex].
- The [tex]\( x \)[/tex]-intercepts of [tex]\( f \)[/tex] become the [tex]\( y \)[/tex]-intercepts of [tex]\( f^{-1} \)[/tex].
- The [tex]\( y \)[/tex]-intercepts of [tex]\( f \)[/tex] become the [tex]\( x \)[/tex]-intercepts of [tex]\( f^{-1} \)[/tex].
Given the original function's characteristics:
- Domain: [tex]\( x \geq -4 \)[/tex]
- Range: [tex]\( y \geq 3 \)[/tex]
- [tex]\( x \)[/tex]-intercept: [tex]\((-1,0)\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((0,8)\)[/tex]
For the inverse function:
- The domain is the range of the original function: [tex]\( y \geq 3 \)[/tex].
- The range is the domain of the original function: [tex]\( x \geq -4 \)[/tex].
- The [tex]\( x \)[/tex]-intercept is the [tex]\( y \)[/tex]-intercept of the original function: [tex]\( (0, 8) \)[/tex].
- The [tex]\( y \)[/tex]-intercept is the [tex]\( x \)[/tex]-intercept of the original function: [tex]\( (-1, 0) \)[/tex].
So, we are looking for a set of characteristics that match:
- Domain: [tex]\( x \geq 3 \)[/tex]
- Range: [tex]\( y \geq -4 \)[/tex]
- [tex]\( x \)[/tex]-intercept: [tex]\((0, 8)\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((-1, 0)\)[/tex]
Comparing this to the given options:
1. Domain: [tex]\( x \geq -3 \)[/tex]; Range: [tex]\( y \geq 4 \)[/tex]; [tex]\( x \)[/tex]-intercept: [tex]\((-8, 0) \)[/tex]; [tex]\( y \)[/tex]-intercept: [tex]\((0, 1) \)[/tex]
This does not match our requirements.
2. Domain: [tex]\( x \geq 4 \)[/tex]; Range: [tex]\( y \geq -3 \)[/tex]; [tex]\( x \)[/tex]-intercept: [tex]\((1, 0) \)[/tex]; [tex]\( y \)[/tex]-intercept: [tex]\((0, -8) \)[/tex]
This option also does not match our requirements.
Therefore, neither of the provided options matches the characteristics of the function's inverse based on the given information.
