**Table of Contents**Show

## Introduction to Functions

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, a function is a rule that assigns a unique output value to each input value. Functions are widely used in various fields such as mathematics, physics, engineering, and computer science.

## Characteristics of Functions

When determining whether a given graph represents a function, there are several key characteristics to consider:

**Domain and Range:**The domain of a function is the set of all possible input values, while the range is the set of all possible output values.**Vertical Line Test:**A graph represents a function if every vertical line intersects the graph at most once.**One-to-One Correspondence:**A function is one-to-one if each input value corresponds to exactly one output value.**Function Notation:**Functions are often represented using function notation, such as f(x) = x^2.

## Graphs of Functions

When looking at graphs, it is important to identify whether the graph represents a function. There are common types of graphs that can help determine whether a graph is a function:

**Linear Functions:**Linear functions have a constant slope and graph as a straight line. They represent functions as long as they pass the vertical line test.**Quadratic Functions:**Quadratic functions have a parabolic shape and represent functions as long as they pass the vertical line test.**Cubic Functions:**Cubic functions have an S-shaped curve and represent functions as long as they pass the vertical line test.**Square Root Functions:**Square root functions have a curved graph and represent functions as long as they pass the vertical line test.**Absolute Value Functions:**Absolute value functions have a V-shape and represent functions as long as they pass the vertical line test.

## Examples of Graphs Representing Functions

Let’s look at some examples of graphs and determine which ones represent functions:

### Example 1: Linear Function

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line with a constant slope. For example, the graph of f(x) = 2x + 1 is a linear function.

In the graph above, every vertical line intersects the graph at most once, indicating that it is a function.

### Example 2: Quadratic Function

A quadratic function has the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola. For example, the graph of f(x) = x^2 is a quadratic function.

In the graph above, every vertical line intersects the graph at most once, indicating that it is a function.

## Identifying Non-Functions

Not all graphs represent functions. There are certain types of graphs that do not satisfy the criteria of a function:

**Vertical Line Test Violation:**If a graph fails the vertical line test and intersects a vertical line at more than one point, it does not represent a function.**Cyclical Graphs:**Cyclical graphs such as circles or spirals do not represent functions as they do not pass the vertical line test.**Disconnected Graphs:**Graphs with disconnected segments do not represent functions as each input does not correspond to a unique output.

## Conclusion

Understanding which graphs represent functions is essential in various mathematical applications. By considering the domain, range, vertical line test, and function notation, one can determine whether a graph is a function. Linear, quadratic, cubic, square root, and absolute value functions are common examples of functions represented by graphs. It is important to be able to identify non-functions based on violations of the vertical line test or disconnected segments. Overall, functions play a crucial role in mathematics and other fields, making it crucial to distinguish which graphs accurately represent functions.