https://file.aiquickdraw.com/mathbot/file/1725506915118imn389mu.jpg

is the function depicted in this image injective surjective or bijective

To determine whether the function shown in the graph is injective, surjective, or bijective, let's analyze the graph carefully.

### Definitions:
1. **Injective (One-to-One):** A function is injective if each input maps to a unique output, i.e., no two different inputs have the same output.
2. **Surjective (Onto):** A function is surjective if every possible output is mapped by at least one input.
3. **Bijective:** A function is bijective if it is both injective and surjective.

### Analysis:
1. **Injective Check:**
   - To be injective, the graph must pass the Horizontal Line Test: every horizontal line should intersect the graph at most once.
   - In this graph, some horizontal lines (especially those around the y-values between -1 and 1) intersect the curve at more than one point. This indicates the function is **not injective**.

2. **Surjective Check:**
   - To be surjective, the function must cover all y-values in its range.
   - The graph appears to cover all y-values from \(-\infty\) to \(\infty\), suggesting it is **surjective**.

3. **Bijective Check:**
   - Since the function is surjective but not injective, it is **not bijective**.

### Conclusion:
- **The function is surjective but not injective, so it is not bijective.**

Would you like a more detailed explanation, or do you have any questions?

### Follow-Up Questions:
1. What is the Horizontal Line Test and why does it determine injectivity?
2. How can you algebraically determine if a function is injective or surjective?
3. What are some examples of functions that are injective, surjective, and bijective?
4. How would transformations affect the injectivity or surjectivity of a function?
5. What implications do these properties have in real-world applications?

### Tip:
When analyzing graphs for injectivity, always check for overlapping outputs with different inputs to determine uniqueness!

Is the function depicted in this image injective, surjective, or bijective?

Analyze the graph of a function to determine if it is injective, surjective, or bijective. Learn about the Horizontal Line Test and how it helps identify injectivity. This problem is suitable for Grades 10-12 students studying functions and graph analysis.

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