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Let's walk through the problem:

Given the function:
\[
f(x) = (x - 8)^2
\]

1. **Finding a domain where \(f\) is one-to-one and non-decreasing**:
   - \(f(x) = (x - 8)^2\) is a quadratic function with its vertex at \(x = 8\). 
   - The function is symmetric about the vertex \(x = 8\). 
   - Quadratic functions like this are increasing on the interval \([8, \infty)\) and decreasing on \((-\infty, 8]\).
   - To make the function **one-to-one** (a requirement for inverse functions), we need to **restrict the domain** to an interval where the function is either strictly increasing or decreasing.

   **Answer** for the first part:  
   - A valid domain is \([8, \infty)\) because the function is non-decreasing (increasing) on this interval.

---

2. **Finding the inverse of \(f\) restricted to this domain**:
   - Start with:
     \[
     y = (x - 8)^2
     \]
   - To solve for \(x\), take the square root of both sides:
     \[
     \sqrt{y} = |x - 8|
     \]
   - Since we are only considering the interval \([8, \infty)\), \(x - 8 \geq 0\), so:
     \[
     \sqrt{y} = x - 8
     \]
   - Solving for \(x\):
     \[
     x = \sqrt{y} + 8
     \]
   - Now, switch \(x\) and \(y\) to get the inverse function:
     \[
     f^{-1}(x) = \sqrt{x} + 8
     \]

   **Answer** for the second part:  
   \[
   f^{-1}(x) = \sqrt{x} + 8
   \]

---

Let me know if you need more details or further clarifications!

Here are five related questions to explore further:
1. What would the domain of the inverse function \(f^{-1}(x) = \sqrt{x} + 8\) be?
2. How does restricting the domain affect the behavior of quadratic functions?
3. How can you verify whether a function is one-to-one?
4. What happens if the domain is restricted to \((-\infty, 8]\) instead?
5. How do you find the inverse of functions that are not quadratic?

**Tip**: Always check the derivative of a function to determine if it is increasing or decreasing on a given interval.

Let f(x) = (x - 8)^2. Find a domain on which f is one-to-one and non-decreasing. Find the inverse of f restricted to this domain.

Learn how to determine the domain where f(x) = (x - 8)^2 is one-to-one and non-decreasing, and find its inverse function. This problem involves quadratic functions, inverse functions, and domain restrictions, suitable for high school students (Grades 9-12).

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