To find the inverse function $f^{-1}(x)$ of $f(x) = \sqrt[4]{7x}$, follow these steps:

1. Express the function in terms of $y$: $y = \sqrt[4]{7x}$

2. Solve for $x$ in terms of $y$: To eliminate the fourth root, raise both sides to the power of 4: $y^4 = 7x$ Now, solve for $x$: $x = \frac{y^4}{7}$

3. Interchange $x$ and $y$: Since we are looking for the inverse function, switch $x$ and $y$: $y = \frac{x^4}{7}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x^4}{7}$

Do you have any questions or want further details?

Here are 5 related questions to explore:

1. How do you verify if a function is one-to-one?
2. What is the significance of finding the inverse of a function?
3. Can all functions have inverses?
4. How do domain restrictions affect the inverse function?
5. How would the process change if the function had a different root or power?

Tip: To check if two functions are truly inverses, apply both compositions $f(f^{-1}(x))$ and $f^{-1}(f(x))$. They should return $x$.