Answer :
To find the characteristics of the inverse function, we need to follow steps based on the properties of inverse functions. Here's a detailed, step-by-step approach:
1. Understanding the given data:
- The original function has the domain [tex]\( x \geq 2 \)[/tex].
- The range of the original function is [tex]\( y \geq -3 \)[/tex].
- The [tex]\( x \)[/tex]-intercept of the original function is [tex]\( (11, 0) \)[/tex].
2. Properties of inverse functions:
- 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.
- The [tex]\( x \)[/tex]-intercept of the original function will correspond to the [tex]\( y \)[/tex]-intercept of the inverse function.
3. Step-by-step conversion for the inverse function:
- The domain of the inverse function is the range of the original function: [tex]\( x \geq -3 \)[/tex].
- The range of the inverse function is the domain of the original function: [tex]\( y \geq 2 \)[/tex].
- The [tex]\( y \)[/tex]-intercept of the inverse function comes from the [tex]\( x \)[/tex]-intercept of the original function being re-interpreted: [tex]\( (0, 11) \)[/tex].
4. Comparing with the given options:
- Option A: domain: [tex]\( x \geq 2 \)[/tex], range: [tex]\( y \geq -3 \)[/tex], [tex]\( x \)[/tex]-intercept: [tex]\( (-11, 0) \)[/tex]
- This retains the original domain and range and changes the [tex]\( x \)[/tex]-intercept incorrectly.
- Option B: domain: [tex]\( x \geq 3 \)[/tex], range: [tex]\( y \geq -2 \)[/tex], [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]
- This changes the domain and range incorrectly.
- Option C: domain: [tex]\( x \geq -3 \)[/tex], range: [tex]\( y \geq 2 \)[/tex], [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]
- This matches the converted domain, range, and correctly identifies the [tex]\( y \)[/tex]-intercept.
- Option D: domain: [tex]\( x \geq -2 \)[/tex], range: [tex]\( y \geq 3 \)[/tex], [tex]\( x \)[/tex]-intercept: [tex]\( (-11, 0) \)[/tex]
- This changes the domain and range incorrectly.
5. Conclusion:
The correct answer is Option C, which has the domain [tex]\( x \geq -3 \)[/tex], range [tex]\( y \geq 2 \)[/tex], and [tex]\( y \)[/tex]-intercept [tex]\( (0, 11) \)[/tex].
Thus, the correct set of information could be characteristics of the function's inverse:
[tex]\[ \text{C. domain: } x \geq -3; \text{ range: } y \geq 2; \text{ y-intercept: } (0, 11). \][/tex]
1. Understanding the given data:
- The original function has the domain [tex]\( x \geq 2 \)[/tex].
- The range of the original function is [tex]\( y \geq -3 \)[/tex].
- The [tex]\( x \)[/tex]-intercept of the original function is [tex]\( (11, 0) \)[/tex].
2. Properties of inverse functions:
- 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.
- The [tex]\( x \)[/tex]-intercept of the original function will correspond to the [tex]\( y \)[/tex]-intercept of the inverse function.
3. Step-by-step conversion for the inverse function:
- The domain of the inverse function is the range of the original function: [tex]\( x \geq -3 \)[/tex].
- The range of the inverse function is the domain of the original function: [tex]\( y \geq 2 \)[/tex].
- The [tex]\( y \)[/tex]-intercept of the inverse function comes from the [tex]\( x \)[/tex]-intercept of the original function being re-interpreted: [tex]\( (0, 11) \)[/tex].
4. Comparing with the given options:
- Option A: domain: [tex]\( x \geq 2 \)[/tex], range: [tex]\( y \geq -3 \)[/tex], [tex]\( x \)[/tex]-intercept: [tex]\( (-11, 0) \)[/tex]
- This retains the original domain and range and changes the [tex]\( x \)[/tex]-intercept incorrectly.
- Option B: domain: [tex]\( x \geq 3 \)[/tex], range: [tex]\( y \geq -2 \)[/tex], [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]
- This changes the domain and range incorrectly.
- Option C: domain: [tex]\( x \geq -3 \)[/tex], range: [tex]\( y \geq 2 \)[/tex], [tex]\( y \)[/tex]-intercept: [tex]\( (0, 11) \)[/tex]
- This matches the converted domain, range, and correctly identifies the [tex]\( y \)[/tex]-intercept.
- Option D: domain: [tex]\( x \geq -2 \)[/tex], range: [tex]\( y \geq 3 \)[/tex], [tex]\( x \)[/tex]-intercept: [tex]\( (-11, 0) \)[/tex]
- This changes the domain and range incorrectly.
5. Conclusion:
The correct answer is Option C, which has the domain [tex]\( x \geq -3 \)[/tex], range [tex]\( y \geq 2 \)[/tex], and [tex]\( y \)[/tex]-intercept [tex]\( (0, 11) \)[/tex].
Thus, the correct set of information could be characteristics of the function's inverse:
[tex]\[ \text{C. domain: } x \geq -3; \text{ range: } y \geq 2; \text{ y-intercept: } (0, 11). \][/tex]
