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To determine which of the graphs represent one-to-one functions, we can apply the **horizontal line test**. A function is one-to-one if and only if every horizontal line crosses the graph at most once.

Now, let’s go through each graph:

1. **Graph a (Top graph)**: This graph appears to be a sinusoidal wave. A horizontal line can intersect this curve at multiple points, so it is **not a one-to-one function**.

2. **Graph b (Second graph)**: The curve represents a vertical ellipse. A horizontal line can intersect this ellipse twice, so it is **not a one-to-one function**.

3. **Graph c (Third graph)**: This is a straight line with a positive slope. A horizontal line will intersect it only once at most, meaning it **is a one-to-one function**.

4. **Graph d (Fourth graph)**: This is a V-shaped function, likely an absolute value function. A horizontal line can intersect the two arms of the "V" at two points, so it is **not a one-to-one function**.

5. **Graph e (Fifth graph)**: This is a parabolic curve opening upwards. A horizontal line will intersect this curve twice, so it is **not a one-to-one function**.

Thus, only **Graph c** represents a one-to-one function.

Would you like further details or have any questions on this?

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Here are five related questions for you to explore:
1. How does the horizontal line test differ from the vertical line test?
2. Why is it important for a function to be one-to-one when determining its inverse?
3. Can a quadratic function ever be a one-to-one function?
4. How do transformations (e.g., shifts or stretches) affect the one-to-one nature of a graph?
5. What is the mathematical definition of an injective (one-to-one) function?

**Tip**: A function is considered one-to-one (injective) if different inputs always produce different outputs.

Select all of the following graphs which are one-to-one functions.

Learn how to determine if a function is one-to-one by applying the horizontal line test. This problem includes multiple graphs, such as sinusoidal, linear, and parabolic, and explores their one-to-one properties. Suitable for students in Grades 9-12.

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