## Introduction

Graphs are an important tool in mathematics, and they are used to represent data and relationships between variables. One of the key concepts in graphing is the idea of a function, which is a set of ordered pairs that relate one variable to another. In this article, we will explore the concept of functions and how to determine if a graph represents a function or not.

## What is a Function?

A function is a set of ordered pairs in which each input has exactly one output. This means that for every value of x, there is only one value of y. The input variable is typically denoted by x, and the output variable is denoted by y. For example, the function f(x) = 2x + 1 can be represented as a set of ordered pairs: {(0,1), (1,3), (2,5), (3,7), …}. In this case, for every value of x, there is only one value of y, which makes it a function.

## How to Determine if a Graph Represents a Function

To determine if a graph represents a function, we can use the vertical line test. The vertical line test is a method of determining if a graph is a function by drawing a vertical line through the graph. If the vertical line intersects the graph at more than one point, then the graph does not represent a function. For example, consider the graph below:

If we draw a vertical line through the graph, we can see that it intersects the graph at two points. Therefore, this graph does not represent a function. On the other hand, consider the graph below:

If we draw a vertical line through the graph, we can see that it intersects the graph at only one point. Therefore, this graph represents a function.

## Examples of Functions and Non-Functions

Let’s look at some more examples of functions and non-functions:

In the first graph, we can see that there is only one y-value for each x-value, which makes it a function. In the second graph, there are two y-values for x=2, which makes it not a function.

In the first graph, there are two x-values for y=3, which makes it not a function. In the second graph, there is only one y-value for each x-value, which makes it a function.

## Conclusion

In conclusion, a function is a set of ordered pairs in which each input has exactly one output. To determine if a graph represents a function, we can use the vertical line test. If the vertical line intersects the graph at more than one point, then the graph does not represent a function. By understanding the concept of functions and how to determine if a graph represents a function or not, we can better interpret and analyze graphs in mathematics.