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Sagot :
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]
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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