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.