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To solve the problem, we need to find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \left( \frac{x}{7} \right)^{\frac{1}{5}} \).

### Step 1: Set up the equation
Start by writing \( y = f(x) \):
\[
y = \left( \frac{x}{7} \right)^{\frac{1}{5}}
\]

### Step 2: Solve for \( x \)
To find the inverse function, we need to solve for \( x \) in terms of \( y \). Begin by isolating \( \frac{x}{7} \):
\[
y^5 = \frac{x}{7}
\]
Now, multiply both sides by 7 to solve for \( x \):
\[
x = 7y^5
\]

### Step 3: Express the inverse function
Now that we have \( x = 7y^5 \), we can write the inverse function by replacing \( y \) with \( x \):
\[
f^{-1}(x) = 7x^5
\]

### Final Answer
The correct option is:
\[
f^{-1}(x) = 7x^5
\]

Would you like further details on how this problem was solved or have any additional questions?

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**Related Questions:**
1. What is the inverse function for \( f(x) = \left(\frac{x}{3}\right)^{\frac{1}{2}} \)?
2. How do you find the inverse of a logarithmic function?
3. What is the process to verify if a function and its inverse are correct?
4. How do transformations affect the inverse of a function?
5. What are the properties of inverse functions?

**Tip:** When finding the inverse of a function, always ensure that you correctly isolate the variable and solve step by step to avoid mistakes.

For the function f(x) = (x/7)^(1/5), find f^(-1)(x).

Explore how to find the inverse function f^(-1)(x) for the given function f(x) = (x/7)^(1/5). This problem includes essential algebraic manipulation techniques and the application of the inverse function theorem. Suitable for students in Grades 11-12.

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