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.

The query submitted by the user: $$ ^3\sqrt{x} + 5 $$ $$ f^{-1} $$

Learn how to find the inverse of the cube root function f(x) = ^3√x + 5. This solution explains the process step by step, using key algebraic principles. Suitable for students in Grades 9-12.

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