https://file.aiquickdraw.com/mathbot/file/1729804651811zxcpvwnb.jpg

To determine whether each function is **one-to-one**, we need to apply the **horizontal line test**. A function is **one-to-one** if any horizontal line intersects its graph at most once.

Let's examine each graph:

1. **Graph 1:**  
   The points are scattered, and multiple points lie on the same horizontal lines (e.g., at \( y = 2 \) and \( y = 1 \)).  
   **Not one-to-one**.

2. **Graph 2:**  
   This graph represents a sinusoidal (wave-like) function. A horizontal line intersects this graph at more than one point in several places.  
   **Not one-to-one**.

3. **Graph 3:**  
   This is a straight line with a positive slope. Any horizontal line will intersect this graph exactly once.  
   **One-to-one**.

4. **Graph 4:**  
   This graph is a straight line with a negative slope. Like Graph 3, any horizontal line will intersect it exactly once.  
   **One-to-one**.

5. **Graph 5:**  
   This is a parabola opening upwards. A horizontal line drawn through the vertex or above/below the vertex will intersect the graph at two points.  
   **Not one-to-one**.

6. **Graph 6:**  
   The points are scattered again, and some horizontal lines intersect multiple points (e.g., at \( y = 3 \)).  
   **Not one-to-one**.

### Summary of one-to-one:
- Graph 1: No
- Graph 2: No
- Graph 3: Yes
- Graph 4: Yes
- Graph 5: No
- Graph 6: No

Do you want further clarification on any graph, or would you like additional details?

Here are 5 related questions you could explore:
1. What is the **horizontal line test**, and why is it used for checking one-to-one functions?
2. How does the concept of **inverse functions** relate to one-to-one functions?
3. Can a function be one-to-one if it has a turning point?
4. Why are straight lines with non-zero slopes always one-to-one?
5. How does being one-to-one affect a function's **domain and range**?

**Tip:** For a function to have an inverse that is also a function, it must pass the horizontal line test, ensuring it is one-to-one!

For each function graphed, state whether it is one-to-one based on the horizontal line test.

Analyze six different graphs to determine whether each function is one-to-one using the horizontal line test. This problem focuses on understanding how the test is applied and why some functions pass or fail. Suitable for students in Grades 10-12.

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