The function $f$ shown in the graph is not invertible because it does not pass the horizontal line test, meaning that for some $y$-values, there are multiple corresponding $x$-values. To make the function invertible, we need to restrict its domain to an interval where it passes the horizontal line test.

Analyzing the options:

• Option A: $0 \leq x \leq 2$
  This interval covers a single arc of the function and passes the horizontal line test, meaning the function is one-to-one within this interval.

• Option B: $0 \leq x \leq 4$
  This interval includes two arcs, so the function fails the horizontal line test here, as it is not one-to-one.

• Option C: $-2 \leq x \leq 2$
  This interval includes a full arc that is symmetric about the y-axis. However, it also passes the horizontal line test, so the function is one-to-one within this interval.

Correct Answers:

• A: $0 \leq x \leq 2$
• C: $-2 \leq x \leq 2$

These intervals would make the function invertible.

Would you like any further clarification or details?

Here are 5 relative questions based on this concept:

1. What does the horizontal line test determine about a function?
2. How do you determine if a function is invertible?
3. How does restricting the domain affect the properties of a function?
4. What is the difference between a one-to-one function and a many-to-one function?
5. How can you graphically determine whether a function is injective?

Tip: When checking if a function is invertible, always apply the horizontal line test to ensure that for every $y$-value, there is only one corresponding $x$-value.