https://file.aiquickdraw.com/mathbot/file/17295592999475eppp2iy.jpg

To determine if a graph represents a **one-to-one function**, we use the **Horizontal Line Test**. If any horizontal line drawn through the graph intersects it more than once, the function is **not one-to-one**. If every horizontal line touches the graph at most once, then the function is **one-to-one**.

Let's analyze each graph:

1. **Graph 1 (Parabola)**:
   - A horizontal line will intersect the graph at two points.
   - **Not one-to-one**.

2. **Graph 2 (Increasing curve)**:
   - A horizontal line will intersect the graph at only one point.
   - **One-to-one**.

3. **Graph 3 (Sine-like wave)**:
   - A horizontal line will intersect the graph at multiple points.
   - **Not one-to-one**.

4. **Graph 4 (Piecewise function with multiple values at certain points)**:
   - Since there are two points on the same vertical line with different y-values, it violates the one-to-one rule.
   - **Not one-to-one**.

5. **Graph 5 (Linear function)**:
   - A horizontal line will intersect the graph at only one point.
   - **One-to-one**.

6. **Graph 6 (Discrete points)**:
   - Since these are discrete points, no horizontal line will intersect more than one point.
   - **One-to-one**.

In summary:
- Graph 1: No
- Graph 2: Yes
- Graph 3: No
- Graph 4: No
- Graph 5: Yes
- Graph 6: Yes

Would you like further explanation or have any questions?

Here are some related questions:
1. How does the vertical line test differ from the horizontal line test?
2. Can a function be one-to-one but not continuous?
3. What does it mean for a function to be bijective?
4. How can we determine if a function is one-to-one algebraically?
5. What role do derivatives play in determining if a function is one-to-one?

**Tip**: When checking graphs for one-to-one functions, always imagine sweeping a horizontal line across the graph to ensure no two points are intersected simultaneously.

For each function graphed below, state whether it is one-to-one.

Analyze six different graphs to determine if they represent one-to-one functions using the Horizontal Line Test. This problem is suitable for high school students and helps in understanding the concept of one-to-one functions and graphical analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation