rite the inverse of b(x) = x5.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter5: Graphs And The Derivative
Section5.CR: Chapter 5 Review
Problem 16CR
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![### Question:
Write the inverse of \( b(x) = x^{\frac{4}{5}} \).
### Your Answer:
- ∘ \( b^{-1}(x) = \sqrt[5]{x^4} \)
- ∘ \( b^{-1}(x) = \log_x \frac{4}{5} \)
- ∘ \( b^{-1}(x) = x^{\frac{5}{4}} \)
---
In this problem, you are asked to find the inverse of the function \( b(x) = x^{\frac{4}{5}} \). Three options for the inverse function \( b^{-1}(x) \) are provided.
The correct approach is to make \(y = b(x)\) and solve for \(x\) in terms of \(y\). This involves the following steps:
1. Set \( y = x^{\frac{4}{5}} \).
2. To solve for \( x \), raise both sides to the reciprocal power of \(\frac{4}{5}\), which is \(\frac{5}{4}\):
\[ x = y^{\frac{5}{4}} \]
3. Therefore, the inverse function is \( b^{-1}(x) = x^{\frac{5}{4}} \).
This analysis helps explain why the third option is the correct answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fece93fa3-d971-4c92-9fbf-77cbc459e71c%2F71ba36ea-b3a4-40aa-809d-a1a2be6599f1%2Fncjm8rp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question:
Write the inverse of \( b(x) = x^{\frac{4}{5}} \).
### Your Answer:
- ∘ \( b^{-1}(x) = \sqrt[5]{x^4} \)
- ∘ \( b^{-1}(x) = \log_x \frac{4}{5} \)
- ∘ \( b^{-1}(x) = x^{\frac{5}{4}} \)
---
In this problem, you are asked to find the inverse of the function \( b(x) = x^{\frac{4}{5}} \). Three options for the inverse function \( b^{-1}(x) \) are provided.
The correct approach is to make \(y = b(x)\) and solve for \(x\) in terms of \(y\). This involves the following steps:
1. Set \( y = x^{\frac{4}{5}} \).
2. To solve for \( x \), raise both sides to the reciprocal power of \(\frac{4}{5}\), which is \(\frac{5}{4}\):
\[ x = y^{\frac{5}{4}} \]
3. Therefore, the inverse function is \( b^{-1}(x) = x^{\frac{5}{4}} \).
This analysis helps explain why the third option is the correct answer.
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