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?

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.