which of the following graphs represents a function

Guri Bee

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20-10-2024

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Unmasking Functions: Decoding Graphs with the Vertical Line Test

In mathematics, a function is a special kind of relationship between inputs and outputs. Every input maps to exactly one output. This seemingly simple definition has a profound visual consequence: the graph of a function must pass the vertical line test.

What is the Vertical Line Test?

The vertical line test is a simple, yet powerful tool for determining if a graph represents a function. Here's how it works:

• Draw a vertical line anywhere on the graph.
• If the vertical line intersects the graph at more than one point, the graph does not represent a function. This is because a single input (the x-coordinate of the vertical line) would correspond to multiple outputs (the y-coordinates of the intersection points).
• If the vertical line intersects the graph at only one point (or not at all) for every vertical line you draw, then the graph represents a function.

Let's illustrate with some examples from GitHub:

Example 1: (from this GitHub repository)

[Image of a graph showing a parabola]

Question: Does the graph represent a function?

Answer: Yes. The vertical line test holds true for all vertical lines drawn. No vertical line intersects the graph more than once.

Example 2: (from this GitHub discussion)

[Image of a graph showing a circle]

Question: Does the graph represent a function?

Answer: No. Notice that any vertical line drawn through the circle will intersect the circle at two points. This violates the vertical line test, indicating the graph does not represent a function.

Understanding the Why

The vertical line test isn't just a handy trick; it's a direct consequence of the function definition. The x-coordinate represents the input, and the y-coordinate represents the output. For a graph to be a function, each input must have only one corresponding output. If a vertical line intersects the graph at multiple points, it means there are multiple outputs for the same input, violating the definition of a function.

Beyond the Test: Function Families

While the vertical line test is a fundamental tool, understanding different types of functions can help you visually identify them. For example:

• Linear Functions: These create straight lines.
• Quadratic Functions: These create parabolas.
• Exponential Functions: These create curves that grow or decay exponentially.

Knowing these families helps you anticipate how a function might behave and whether it will pass the vertical line test.

Conclusion

The vertical line test provides a simple, visual way to determine if a graph represents a function. By understanding the connection between the test and the definition of a function, and by familiarizing yourself with common function families, you can gain valuable insights into the nature of mathematical relationships.