To make the function $f(x) = (x + 3)^2$ one-to-one and still maintain the same range, we need to restrict its domain.

The function $f(x) = (x + 3)^2$ is a parabola that opens upward, with its vertex at $x = -3$. The function is symmetric around this vertex, so if we choose a domain where $x \geq -3$, the function will increase from its minimum point without repeating any values, making it one-to-one.

Solution:

To make $f(x)$ one-to-one:

• Restrict the domain to $x \geq -3$.

This restriction ensures the function remains increasing from $x = -3$ onward, thus maintaining a unique output for each input.

Would you like more details on this process or have any questions?

Here are 5 questions that extend this concept:

1. Why does restricting the domain to $x \geq -3$ make the function one-to-one?
2. How would the range of the function be affected if we used $x \leq -3$ instead?
3. What other types of functions require domain restrictions to be one-to-one?
4. How does the concept of an inverse function relate to making a function one-to-one?
5. Could we use any other value besides $x = -3$ to split the domain and make the function one-to-one?

Tip: For parabolas, restricting the domain to one side of the vertex (either left or right) is a common way to make the function one-to-one.