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Select the correct answer.

The domain, range, and [tex]$x$[/tex]-intercept of a one-to-one function are shown.
\begin{tabular}{|l|l|c|}
\hline \multicolumn{1}{|c|}{ domain: } & \multicolumn{1}{|c|}{ range: } & [tex]$x$[/tex]-intercept: \\
\hline [tex]$x \geq 2$[/tex] & [tex]$y \geq -3$[/tex] & [tex]$(11,0)$[/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]$y \geq 2$[/tex]; [tex]$y$[/tex]-intercept: [tex]$(0,11)$[/tex]

B. domain: [tex]$x \geq 3$[/tex]; range: [tex]$y \geq -2$[/tex]; [tex]$y$[/tex]-intercept [tex]$(0,11)$[/tex]

C. domain: [tex]$x \geq -2$[/tex]; range: [tex]$y \geq 3$[/tex]; [tex]$x$[/tex]-intercept: [tex]$(-11,0)$[/tex]

D. domain: [tex]$x \geq 2$[/tex]; range: [tex]$y \geq -3$[/tex]; [tex]$x$[/tex]-intercept: [tex]$(-11,0)$[/tex]



Answer :

GinnyAnswer
To solve this problem, let's analyze the properties and characteristics of a one-to-one function and its inverse.

1. Domain and Range:
- The domain of the original function becomes the range of the inverse function.
- The range of the original function becomes the domain of the inverse function.

2. Intercepts:
- The [tex]\( x \)[/tex]-intercept of the original function, say [tex]\((a, 0)\)[/tex], becomes the [tex]\( y \)[/tex]-intercept of the inverse function, which will be [tex]\((0, a)\)[/tex].

Given the original function:
- Domain: [tex]\( x \geq 2 \)[/tex]
- Range: [tex]\( y \geq -3 \)[/tex]
- [tex]\( x \)[/tex]-intercept: [tex]\( (11, 0) \)[/tex]

Let's find the corresponding properties for the inverse function:
- Domain of the inverse function: This will be the range of the original function, so we have [tex]\( x \geq -3 \)[/tex].
- Range of the inverse function: This will be the domain of the original function, so we have [tex]\( y \geq 2 \)[/tex].
- [tex]\( y \)[/tex]-intercept: The [tex]\( x \)[/tex]-intercept [tex]\( (11, 0) \)[/tex] of the original function becomes the [tex]\( y \)[/tex]-intercept of the inverse function, so we have [tex]\( (0, 11) \)[/tex].

Now, let's match these properties with the given answer choices:

A. Domain: [tex]\( x \geq -3 \)[/tex]; Range: [tex]\( y \geq 2 \)[/tex]; [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]
This matches our calculated properties for the inverse function.

B. Domain: [tex]\( x \geq 3 \)[/tex]; Range: [tex]\( y \geq -2 \)[/tex]; [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]
This does not match our calculated properties.

C. Domain: [tex]\( x \geq -2 \)[/tex]; Range: [tex]\( y \geq 3 \)[/tex]; [tex]\( x \)[/tex]-intercept: [tex]\( (-11, 0) \)[/tex]
This does not match our calculated properties.

D. Domain: [tex]\( x \geq 2 \)[/tex]; Range: [tex]\( y \geq -3 \)[/tex]; [tex]\( x \)[/tex]-intercept: [tex]\( (-11, 0) \)[/tex]
This does not match our calculated properties.

Therefore, the correct answer is:

[tex]\[ \boxed{A} \][/tex]

The inverse function's characteristics are:
- Domain: [tex]\( x \geq -3 \)[/tex]
- Range: [tex]\( y \geq 2 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]

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