We are tasked with finding the inverse function $f^{-1}(x)$ for the function $f(x) = \sqrt[3]{5x + 4}$.

Steps to Find the Inverse:

1. Express the function with $y$: $y = f(x) = \sqrt[3]{5x + 4}$

2. Swap $y$ and $x$ to find the inverse: $x = \sqrt[3]{5y + 4}$

3. Solve for $y$:

   • Cube both sides to eliminate the cube root: $x^3 = 5y + 4$
   • Solve for $y$: $5y = x^3 - 4$ $y = \frac{x^3 - 4}{5}$

4. Write the inverse function: $f^{-1}(x) = \frac{x^3 - 4}{5}$

Final Answer:

The inverse function is: $f^{-1}(x) = \frac{x^3 - 4}{5}$

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Would you like more details on this process, or do you have any specific questions about the inverse function?

Here are some related questions:

1. How do you verify that $f$ and $f^{-1}$ are true inverses?
2. What would happen if the original function were a square root function instead of a cube root?
3. How do you find the domain and range of an inverse function?
4. How can you graph both $f(x)$ and $f^{-1}(x)$ on the same plot?
5. What are the steps to finding the inverse of a function when the function is a polynomial?

Tip: When finding the inverse, always remember to switch $x$ and $y$, then solve for $y$.