Each input has exactly one output

Oct 13, 2025 by miodeya.com 22 views

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function is a special type of relation where each input is associated with exactly one output. In this article, we will examine four given relations and determine which ones are functions.

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between two sets of values. A function is often denoted by a symbol, such as f(x), where x is the input and f(x) is the output.

Properties of a Function

A function must satisfy the following properties:

Each input has exactly one output: This means that for every input, there is only one corresponding output.
The output is determined by the input: This means that the output is dependent on the input, and not on any other factor.
The function is well-defined: This means that the function is defined for all possible inputs, and there are no gaps or overlaps in the output.

Examining the Relations

Now, let's examine the four given relations and determine which ones are functions.

Relation 1: ${$ (4,4),(5,6),(6,8),(6,10),(7,12)$}$

This relation is not a function because it does not satisfy the property of each input having exactly one output. The input 6 is associated with two different outputs, 8 and 10.

Relation 2: ${$ (6,9),(7,19),(8,29),(8,39),(9,49)$}$

This relation is not a function because it does not satisfy the property of each input having exactly one output. The input 8 is associated with two different outputs, 29 and 39.

Relation 3: ${$ (4,4),(5,4),(6,4),(7,4),(8,4)$}$

This relation is a function because it satisfies the property of each input having exactly one output. Each input is associated with the output 4.

Relation 4: ${$ (1,2),(2,3),(3,4),(4,5),(5,6)$}$

This relation is a function because it satisfies the property of each input having exactly one output. Each input is associated with a unique output.

Conclusion

In conclusion, a function is a relation between a set of inputs and a set of possible outputs, where each input is associated with exactly one output. The four given relations were examined, and it was determined that only Relations 3 and 4 are functions.

Key Takeaways

A function is a relation between a set of inputs and a set of possible outputs.
A function must satisfy the property of each input having exactly one output.
The output of a function is determined by the input.
A function is well-defined for all possible inputs.

References

In the previous article, we discussed the concept of functions and relations, and examined four given relations to determine which ones are functions. In this article, we will answer some frequently asked questions about functions and relations.

Q: What is the difference between a function and a relation?

A: A relation is a set of ordered pairs, where each pair consists of an input and an output. A function is a special type of relation where each input is associated with exactly one output.

Q: What are the properties of a function?

A: A function must satisfy the following properties:

Each input has exactly one output: This means that for every input, there is only one corresponding output.
The output is determined by the input: This means that the output is dependent on the input, and not on any other factor.
The function is well-defined: This means that the function is defined for all possible inputs, and there are no gaps or overlaps in the output.

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you need to check if each input is associated with exactly one output. If there are any inputs that are associated with multiple outputs, then the relation is not a function.

Q: Can a function have multiple outputs for the same input?

A: No, a function cannot have multiple outputs for the same input. If a function has multiple outputs for the same input, then it is not a function.

Q: Can a function have no outputs for a particular input?

A: Yes, a function can have no outputs for a particular input. This is known as a "hole" in the function.

Q: Can a function have multiple inputs that map to the same output?

A: Yes, a function can have multiple inputs that map to the same output. This is known as a "multiple" or "many-to-one" function.

Q: Can a function have a single input that maps to multiple outputs?

A: No, a function cannot have a single input that maps to multiple outputs. If a function has a single input that maps to multiple outputs, then it is not a function.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on a coordinate plane, where the x-axis represents the input and the y-axis represents the output.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible inputs.

Q: What is the range of a function?

A: The range of a function is the set of all possible outputs.

Q: Can a function have an empty domain?

A: Yes, a function can have an empty domain. This means that there are no possible inputs.

Q: Can a function have an empty range?

A: Yes, a function can have an empty range. This means that there are no possible outputs.

Conclusion

In conclusion, functions and relations are important concepts in mathematics, and understanding the properties and characteristics of functions is essential for solving problems and graphing functions. We hope that this article has helped to answer some of the frequently asked questions about functions and relations.

Key Takeaways

A function is a special type of relation where each input is associated with exactly one output.
A function must satisfy the property of each input having exactly one output.
The output of a function is determined by the input.
A function is well-defined for all possible inputs.
A function can have multiple inputs that map to the same output.
A function can have a single input that maps to multiple outputs.
A function can have an empty domain or an empty range.