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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]

Sagot :

GinnyAnswer
To determine the characteristics of the inverse function, we need to understand the relationships between the domain, range, and intercepts of a 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 becomes the [tex]$y$[/tex]-intercept of the inverse function, because the inverse function swaps the roles of [tex]$x$[/tex] and [tex]$y$[/tex].

Given:
- The domain of the function is [tex]\( x \geq 2 \)[/tex].
- The range of the function is [tex]\( y \geq -3 \)[/tex].
- The [tex]\( x \)[/tex]-intercept of the original function is [tex]\((11, 0)\)[/tex].

Now let's find the corresponding characteristics of the inverse function:

1. Domain of the Inverse Function: Since the range of the original function is [tex]\( y \geq -3 \)[/tex], this will become the domain of the inverse function. So the domain of the inverse function is [tex]\( x \geq -3 \)[/tex].

2. Range of the Inverse Function: Since the domain of the original function is [tex]\( x \geq 2 \)[/tex], this will become the range of the inverse function. So the range of the inverse function is [tex]\( y \geq 2 \)[/tex].

3. Intercept of the Inverse Function: The [tex]$x$[/tex]-intercept of the original function is [tex]\((11, 0)\)[/tex]. This point will be swapped to become the [tex]$y$[/tex]-intercept of the inverse function. Thus, the [tex]$y$[/tex]-intercept of the inverse function is [tex]\((0, 11)\)[/tex].

Summarizing the characteristics of the inverse function:
- Domain: [tex]\( x \geq -3 \)[/tex]
- Range: [tex]\( y \geq 2 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((0, 11)\)[/tex]

Upon examining the given options, we see that Option A matches all these characteristics:
- Domain: [tex]\( x \geq -3 \)[/tex]
- Range: [tex]\( y \geq 2 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((0, 11)\)[/tex]

Therefore, the correct answer is:

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