In mathematics, 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. To determine if a given relation is a function, there are various ways to test it. In this article, we will explore different scenarios to determine which of the following represent a function.

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## Defining a Function

A function is typically denoted by a rule or expression that maps each input value to a unique output value. Formally, a function f from a set A to a set B is defined as a rule that assigns to each element in set A exactly one element in set B. It is important to note that in a function, each input must have only one corresponding output.

## Key Characteristics of a Function

There are several key characteristics that a function must satisfy:

**Domain and Range:**The domain of a function is the set of all possible input values, and the range is the set of all possible output values.**One-to-One Mapping:**Each input value must have exactly one unique output value.**Vertical Line Test:**If a vertical line intersects the graph of a relation in more than one point, then the relation is not a function.

## Testing for Functions

There are several methods to test if a given relation represents a function:

### Mapping Diagram

A mapping diagram is a visual representation of a relation that shows how each input is related to an output. If each input has only one arrow pointing to an output, then the relation is a function.

### Table of Values

Creating a table of values is another way to determine if a relation is a function. If each input value corresponds to only one output value in the table, then it is a function.

### Vertical Line Test

The vertical line test is a graphical method to determine if a relation represents a function. If a vertical line intersects the graph of the relation at more than one point, then it is not a function.

### Algebraic Method

When given an algebraic expression, you can determine if it represents a function by solving for the output variable. If each input value corresponds to only one output value, then it is a function.

## Examples of Functions

Let’s consider some examples to illustrate which of the following represent a function:

### Example 1:

**Relation:** {(1,2), (2,4), (3,6), (4,8)}

**Represent a Function:** Yes, each input value has exactly one corresponding output value.

### Example 2:

**Relation:** {(1,2), (1,3), (2,4), (3,6)}

**Represent a Function:** No, the input value of 1 has two different output values (2 and 3).

### Example 3:

**Relation:** {(1,2), (2,4), (3,2)}

**Represent a Function:** Yes, each input value has only one output value, even if two inputs have the same output.

## Conclusion

In conclusion, a function is a relation where each input value corresponds to exactly one output value. Various methods such as mapping diagrams, tables of values, the vertical line test, and algebraic methods can be used to determine if a given relation represents a function. It is important to understand the key characteristics of a function and apply the appropriate tests to determine whether a relation is a function.