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The question in the image asks the following:

1. Let \( f(x) = (x - 1)^2 \). 
   - Find a domain where \( f \) is one-to-one and non-decreasing.
   - Find the inverse of \( f \) restricted to this domain.

### Step 1: Finding a Domain Where \( f(x) \) is One-to-One and Non-Decreasing

The function \( f(x) = (x - 1)^2 \) is a parabola, which opens upwards (since the coefficient of \( x^2 \) is positive). A parabola is not one-to-one over its entire domain because it is symmetric around its vertex. The vertex of \( f(x) = (x - 1)^2 \) is at \( x = 1 \).

To make the function one-to-one, we need to restrict the domain to either the left or right of the vertex. Since the question asks for a domain where the function is non-decreasing, we choose the interval where \( f(x) \) is increasing. This happens for \( x \geq 1 \).

Thus, the domain on which \( f(x) \) is one-to-one and non-decreasing is:
\[
x \geq 1
\]

### Step 2: Finding the Inverse of \( f(x) \) on the Restricted Domain

To find the inverse, we start with the equation for \( f(x) \):
\[
y = (x - 1)^2
\]
Now, solve for \( x \) in terms of \( y \):
\[
\sqrt{y} = x - 1
\]
\[
x = \sqrt{y} + 1
\]
Since the domain is restricted to \( x \geq 1 \), we take the positive square root.

Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt{x} + 1
\]

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

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Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How do we determine the vertex of a quadratic function?
2. Why is a function not one-to-one over its entire domain if it is symmetric?
3. Can the domain restriction be \( x \leq 1 \) instead, and if so, how would the inverse change?
4. What is the general method for finding the inverse of a function?
5. How does restricting the domain of a function affect its graph?

**Tip:** When finding the inverse of a quadratic function, always be mindful of domain restrictions, as a quadratic function is not naturally one-to-one.

Let f(x) = (x - 1)^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 - 1)^2 is one-to-one and non-decreasing, and how to find the inverse of the function on that restricted domain. This problem involves quadratic functions, domain restrictions, and inverse functions, suitable for high school students.

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