To find the inverse of the function $f(x) = x^3 - 6$, we need to follow these steps:

Step 1: Replace $f(x)$ with $y$

Let $y = f(x)$, so: $y = x^3 - 6$

Step 2: Swap $x$ and $y$

To find the inverse, swap $x$ and $y$: $x = y^3 - 6$

Step 3: Solve for $y$

Now solve for $y$ in terms of $x$:

$x + 6 = y^3$ $y = \sqrt[3]{x + 6}$

Step 4: Write the inverse function

So the inverse function is: $f^{-1}(x) = \sqrt[3]{x + 6}$

Final Answer:

$f^{-1}(x) = \sqrt[3]{x + 6}$

Let me know if you would like further clarification on any steps, or details on how the function and its inverse relate!

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Here are five related questions you might find interesting:

1. How can you verify that $f(x) = x^3 - 6$ is one-to-one?
2. What is the domain and range of $f(x)$ and $f^{-1}(x)$?
3. How do you check if two functions are inverses of each other?
4. Can the inverse function $f^{-1}(x)$ have any restrictions in its domain?
5. How would this process change for more complicated functions like $f(x) = x^3 - 6x + 2$?

Tip: When finding the inverse, remember that the graph of a function and its inverse are reflections of each other across the line $y = x$.