The given function is $f(x) = \sqrt[3]{x^5 - 3}$, and we are asked to find its inverse, $f^{-1}(x)$.

To find the inverse function, we need to follow these steps:

1. Express the function equation:
   $y = \sqrt[3]{x^5 - 3}$

2. Swap $x$ and $y$:
   $x = \sqrt[3]{y^5 - 3}$

3. Solve for $y$:
   First, eliminate the cube root by cubing both sides:
   $x^3 = y^5 - 3$

4. Isolate $y^5$:
   $y^5 = x^3 + 3$

5. Solve for $y$:
   Now take the fifth root of both sides:
   $y = \sqrt[5]{x^3 + 3}$

Thus, the inverse function is:

$f^{-1}(x) = \sqrt[5]{x^3 + 3}$

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Would you like further details on how to solve this or have any other questions? Here are some related topics you could explore:

1. How to check if a function and its inverse are correct.
2. Inverse functions for other types of operations (e.g., quadratic, exponential).
3. What does it mean geometrically for a function to have an inverse?
4. The domain and range of the inverse function.
5. How to graph a function and its inverse.

Tip: Remember, to find the inverse, the function must be one-to-one (each $x$-value maps to exactly one $y$-value), which is a requirement for the existence of an inverse.