It seems you are asking for the inverse of a function involving the cube root of $x$. Let's first define the function and then find its inverse step by step.

Step 1: Define the function

Given: $f(x) = ^3\sqrt{x} + 5$ This can be rewritten as: $f(x) = x^{\frac{1}{3}} + 5$

Step 2: Swap $f(x)$ and $x$

To find the inverse, swap $f(x)$ with $x$ and solve for $y$: $x = y^{\frac{1}{3}} + 5$

Step 3: Solve for $y$

Now, subtract 5 from both sides: $x - 5 = y^{\frac{1}{3}}$

Next, cube both sides to get rid of the cube root: $(x - 5)^3 = y$

Step 4: Express the inverse

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

Final Answer:

$f^{-1}(x) = (x - 5)^3$

Would you like further details or have any questions about this?

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Here are five related questions for further exploration:

1. What is the domain and range of the original function $f(x) = ^3\sqrt{x} + 5$?
2. How does the graph of the original function compare to the graph of its inverse?
3. What would change if we had $f(x) = ^n\sqrt{x} + 5$ for some other value of $n$?
4. Can you generalize the process of finding inverses for any root-based function?
5. How do you check if two functions are inverses of each other?

Tip: When finding the inverse of a function, always ensure the original function is one-to-one (passes the horizontal line test) to guarantee that the inverse is valid.