The question is asking which restrictions on the domain make the function invertible. A function is invertible if it is one-to-one (i.e., each $y$-value is mapped from a unique $x$-value), meaning it must pass the horizontal line test.

From the graph provided, the function is not one-to-one over its entire domain because it fails the horizontal line test; there are multiple $x$-values for some $y$-values. However, by restricting the domain to specific intervals, we can make it one-to-one.

Evaluating Each Interval

1. $-3 \leq x \leq 9$: This interval does not make the function one-to-one, as there are sections within this interval where a horizontal line intersects the curve more than once.

2. $x \geq 9$: This interval isolates a part of the curve where the function appears to be strictly decreasing, making it one-to-one.

3. $x < 4$: This interval includes parts of the curve where the function is not one-to-one, so it does not work.

4. $4 \leq x \leq 9$: This interval appears to make the function one-to-one since the function is either strictly increasing or decreasing within this interval.

5. $x \geq 4$: This interval does not work, as the function is not one-to-one when $x \geq 4$ due to multiple points sharing the same $y$-values.

6. $x \leq -3$: This interval restricts the domain to a section of the curve that is strictly increasing, making the function one-to-one.

7. $-3 \leq x \leq 4$: This interval fails the horizontal line test due to multiple $x$-values mapping to the same $y$-values.

Conclusion

The selections that make the function invertible are:

• $x \geq 9$
• $4 \leq x \leq 9$
• $x \leq -3$

These choices ensure that the function behaves in a strictly monotonic (increasing or decreasing) manner, allowing it to be inverted.

Would you like more details or explanations?

Here are some follow-up questions for further exploration:

1. What does it mean for a function to pass the horizontal line test?
2. How do we determine if a function is one-to-one over a specific interval?
3. Can a function be invertible on different intervals even if it's not invertible over its entire domain?
4. What are the implications of restricting a function’s domain in real-world applications?
5. How does the concept of function inverses apply in calculus and algebra?

Tip: When checking if a function is invertible on an interval, look for sections where the function is strictly increasing or strictly decreasing.