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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.

To which intervals could we restrict the domain of function f to make it an invertible function?

Learn how to restrict the domain of a function to make it invertible by applying the horizontal line test. This problem focuses on analyzing specific intervals and determining where the function becomes one-to-one. Suitable for high school students in Grades 9-12.

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