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The given function is \( f(x) = (x + 2)^2 \).

### Part 1: Find a domain where \( f(x) \) is one-to-one and non-decreasing.
- The function \( f(x) = (x + 2)^2 \) is a parabola, which is symmetric about \( x = -2 \). 
- Parabolas are decreasing to the left of the vertex and increasing to the right. So, to make the function one-to-one, we need to restrict the domain to where the function is either increasing or decreasing.
- To find a non-decreasing domain, we choose the portion of the function where it increases, which happens to the right of the vertex \( x = -2 \). So the domain should be \( x \geq -2 \).

Thus, the restricted domain is \( x \geq -2 \).

### Part 2: Find the inverse of \( f(x) \) on this restricted domain.
- To find the inverse, first set \( y = f(x) = (x + 2)^2 \).
- Solve for \( x \) in terms of \( y \):
  \[
  y = (x + 2)^2
  \]
  Take the square root of both sides:
  \[
  \sqrt{y} = x + 2
  \]
  Since we are restricting the domain to \( x \geq -2 \), we take the positive square root:
  \[
  x = \sqrt{y} - 2
  \]
- Therefore, the inverse function is:
  \[
  f^{-1}(x) = \sqrt{x} - 2
  \]

### Final Answer:
1. The domain where \( f(x) \) is one-to-one and non-decreasing: \( x \geq -2 \).
2. The inverse of \( f(x) \) on this domain: \( f^{-1}(x) = \sqrt{x} - 2 \).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you verify that a function is one-to-one on a given domain?
2. Why is it necessary to restrict the domain of quadratic functions to find an inverse?
3. What happens if we restrict the domain to \( x \leq -2 \) instead?
4. How can you graphically interpret the inverse of a function?
5. How would you compute the derivative of the inverse function?

**Tip**: Always verify that a function is one-to-one on the chosen domain before finding its inverse, as functions need to be bijective to have an inverse.

Let f(x) = (x + 2)^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 find the domain where the quadratic function f(x) = (x + 2)^2 is one-to-one and non-decreasing. Discover how to calculate the inverse of f(x) on this restricted domain. This problem covers quadratic functions, domain restrictions, and inverse functions, suitable for students in Grades 10-12.

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