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The domain, range, and intercepts of a function are shown.

\begin{tabular}{|l|l|}
\hline Domain: [tex]$x \geq -4$[/tex] & [tex]$x$[/tex]-intercept: [tex]$(-1,0)$[/tex] \\
\hline Range: [tex]$y \geq 3$[/tex] & [tex]$y$[/tex]-intercept: [tex]$(0,8)$[/tex] \\
\hline
\end{tabular}

Which set of information could be characteristics of the function's inverse?

A. 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]

B. 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]

Sagot :

GinnyAnswer
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.
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