Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Let's solve the given question step-by-step.
### Part (a): Finding the Inverse Function
We are given the function:
[tex]\[ f(x) = x + 15 \][/tex]
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x + 15 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse equation:
[tex]\[ x = y + 15 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = x - 15 \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = x - 15 \][/tex]
Therefore, the correct choice is:
[tex]\[ D. \; f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
### Part (b): Verifying the Inverse Function
We will now verify that the inverse function [tex]\( f^{-1}(x) = x - 15 \)[/tex] is correct by showing that:
1. [tex]\( f(f^{-1}(x)) = x \)[/tex]
2. [tex]\( f^{-1}(f(x)) = x \)[/tex]
1. Verification of [tex]\( f(f^{-1}(x)) = x \)[/tex]:
- Substitute [tex]\( f^{-1}(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f(x - 15) \][/tex]
- Apply the function [tex]\( f \)[/tex]:
[tex]\[ f(x - 15) = (x - 15) + 15 = x \][/tex]
Therefore,
[tex]\[ f(f^{-1}(x)) = x \][/tex]
2. Verification of [tex]\( f^{-1}(f(x)) = x \)[/tex]:
- Substitute [tex]\( f(x) \)[/tex] into [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(x + 15) \][/tex]
- Apply the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(x + 15) = (x + 15) - 15 = x \][/tex]
Therefore,
[tex]\[ f^{-1}(f(x)) = x \][/tex]
### Conclusion
Since both verifications hold true, we have confirmed that the inverse function is indeed correct. Thus, the inverse function is:
[tex]\[ f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
And the correct choice is:
[tex]\[ D. \; f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
### Part (a): Finding the Inverse Function
We are given the function:
[tex]\[ f(x) = x + 15 \][/tex]
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x + 15 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse equation:
[tex]\[ x = y + 15 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = x - 15 \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = x - 15 \][/tex]
Therefore, the correct choice is:
[tex]\[ D. \; f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
### Part (b): Verifying the Inverse Function
We will now verify that the inverse function [tex]\( f^{-1}(x) = x - 15 \)[/tex] is correct by showing that:
1. [tex]\( f(f^{-1}(x)) = x \)[/tex]
2. [tex]\( f^{-1}(f(x)) = x \)[/tex]
1. Verification of [tex]\( f(f^{-1}(x)) = x \)[/tex]:
- Substitute [tex]\( f^{-1}(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f(x - 15) \][/tex]
- Apply the function [tex]\( f \)[/tex]:
[tex]\[ f(x - 15) = (x - 15) + 15 = x \][/tex]
Therefore,
[tex]\[ f(f^{-1}(x)) = x \][/tex]
2. Verification of [tex]\( f^{-1}(f(x)) = x \)[/tex]:
- Substitute [tex]\( f(x) \)[/tex] into [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(x + 15) \][/tex]
- Apply the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(x + 15) = (x + 15) - 15 = x \][/tex]
Therefore,
[tex]\[ f^{-1}(f(x)) = x \][/tex]
### Conclusion
Since both verifications hold true, we have confirmed that the inverse function is indeed correct. Thus, the inverse function is:
[tex]\[ f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
And the correct choice is:
[tex]\[ D. \; f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
