At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
