https://file.aiquickdraw.com/mathbot/file/1725480973491xzie9e0h.jpg

To solve for the inverse of the function \( f(x) = \sqrt[4]{\frac{x^5}{4}} \), let's follow these steps:

1. **Express the function**: \( f(x) = \sqrt[4]{\frac{x^5}{4}} \).

2. **Rewrite the equation** to find the inverse:
   \[
   y = \sqrt[4]{\frac{x^5}{4}}
   \]

3. **Isolate \(x\)** by raising both sides to the fourth power:
   \[
   y^4 = \frac{x^5}{4}
   \]

4. **Solve for \(x\)**:
   \[
   x^5 = 4y^4
   \]
   \[
   x = \left(4y^4\right)^{\frac{1}{5}}
   \]

5. **Express the inverse function**:
   \[
   f^{-1}(x) = \left(4x^4\right)^{\frac{1}{5}} = 4^{\frac{1}{5}} \cdot x^{\frac{4}{5}}
   \]

6. **Match the solution with the given choices**:
   \[
   f^{-1}(x) = 4^{\frac{1}{5}} x^{\frac{4}{5}}
   \]

Given the answer choices in the image, this matches with the answer \( f^{-1}(x) = \left(\frac{x^4}{4}\right)^{\frac{1}{5}} \).

This corresponds to the answer **option (A):** \( f^{-1}(x) = \left(\frac{x^4}{4}\right)^{\frac{1}{5}} \).

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Do you want more details on any of these steps or have any questions?

Here are some related questions for further practice:
1. How do you find the inverse of a square root function?
2. What is the general process to find the inverse of any given function?
3. Can you determine the domain and range of the inverse function you found?
4. What happens if the original function \( f(x) \) was different, such as \( f(x) = \sqrt{\frac{x^2}{2}} \)?
5. How do you verify if two functions are inverses of each other?

**Tip:** When finding the inverse of a function, it's important to ensure that the operations you apply to isolate \( x \) are reversible and maintain the function's domain and range.

For the function \( f(x) = \sqrt[4]{\frac{x^5}{4}} \), find \( f^{-1}(x) \).

Learn how to find the inverse of the function \( f(x) = \sqrt[4]{\frac{x^5}{4}} \) with a step-by-step solution. This problem involves understanding inverse functions, algebraic manipulation, and radicals. Suitable for advanced high school students in Grades 11-12.

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