How to Determine if a Relation is a Function

Oct 13, 2025 by miodeya.com 24 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. It is a way of assigning each input to exactly one output. In this article, we will explore three sets of ordered pairs and determine which one is not a function.

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 assigning each input to exactly one output. In other words, for every input, there is only one corresponding output. This is the key characteristic of a function.

Set A: A Function or Not?

Let's examine the first set of ordered pairs:

A = {(-5,-7), (-4,-6), (-3,-5), (-2,-4), (-1,-3)}

In this set, each input is assigned to exactly one output. For example, the input -5 is assigned to the output -7, and the input -4 is assigned to the output -6. This is a clear example of a function, as each input has a unique output.

Set B: A Function or Not?

Now, let's examine the second set of ordered pairs:

B = {(-4,-2), (-1,-1), (1,3), (1,4), (7,10)}

At first glance, this set appears to be a function, as each input is assigned to a unique output. However, upon closer inspection, we notice that the input 1 is assigned to two different outputs: 3 and 4. This is a problem, as a function requires each input to have a unique output.

Set C: A Function or Not?

Unfortunately, we are not given a set of ordered pairs for Set C. However, we can still discuss the characteristics of a function and how to determine if a relation is a function.

How to Determine if a Relation is a Function

To determine if a relation is a function, we need to check if each input is assigned to a unique output. In other words, we need to check if there are any duplicate outputs for a given input. If there are, then the relation is not a function.

Conclusion

In conclusion, the set of ordered pairs that is not a function is Set B. This is because the input 1 is assigned to two different outputs: 3 and 4. This violates the key characteristic of a function, which requires each input to have a unique output.

Key Takeaways

A function is a relation between a set of inputs and a set of possible outputs.
Each input in a function must be assigned to a unique output.
If there are any duplicate outputs for a given input, then the relation is not a function.

Real-World Applications

Functions are used in many real-world applications, such as:

Mathematics: Functions are used to model real-world phenomena, such as population growth and chemical reactions.
Computer Science: Functions are used to write efficient and modular code, making it easier to understand and maintain.
Economics: Functions are used to model economic systems, such as supply and demand curves.

Final Thoughts

In conclusion, functions are an important concept in mathematics and have many real-world applications. By understanding what a function is and how to determine if a relation is a function, we can better appreciate the beauty and power of mathematics.

References

[1] "Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Functions in Real-World Applications" by Wolfram MathWorld

Glossary

Domain: The set of all possible inputs for a function.
Range: The set of all possible outputs for a function.
Function: A relation between a set of inputs and a set of possible outputs, where each input is assigned to a unique output.
Relation: A set of ordered pairs, where each pair consists of an input and an output.
Q&A: Functions and Relations =============================

In our previous article, we discussed the concept of functions and relations, and how to determine if a relation is a function. 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 function is a relation between a set of inputs and a set of possible outputs, where each input is assigned to a unique output. A relation, on the other hand, is a set of ordered pairs, where each pair consists of an input and an output. Not all relations are functions, but all functions are relations.

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 assigned to a unique output. In other words, you need to check if there are any duplicate outputs for a given input. If there are, then the relation is not a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible inputs for the function. In other words, it is the set of all values that can be plugged into the function.

Q: What is the range of a function?

A: The range of a function is the set of all possible outputs for the function. In other words, it is the set of all values that the function can produce.

Q: Can a function have a domain with more than one element?

A: Yes, a function can have a domain with more than one element. For example, the function f(x) = x^2 has a domain of all real numbers, which is an infinite set.

Q: Can a function have a range with more than one element?

A: Yes, a function can have a range with more than one element. For example, the function f(x) = x^2 has a range of all non-negative real numbers, which is an infinite set.

Q: Can a function be one-to-one?

A: Yes, a function can be one-to-one. A one-to-one function is a function where each output corresponds to exactly one input. In other words, it is a function where each output is unique.

Q: Can a function be onto?

A: Yes, a function can be onto. An onto function is a function where each output corresponds to at least one input. In other words, it is a function where each output is reached.

Q: Can a function be both one-to-one and onto?

A: Yes, a function can be both one-to-one and onto. Such a function is called a bijection.

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

A: A function is a relation between a set of inputs and a set of possible outputs, where each input is assigned to a unique output. A bijection is a function that is both one-to-one and onto.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to swap the x and y values of the function. In other words, you need to solve the equation y = f(x) for x.

Q: What is the inverse of a function?

A: The inverse of a function is a function that undoes the original function. In other words, it is a function that takes the output of the original function and returns the input.

Q: Can a function have an inverse?

A: Yes, a function can have an inverse. However, not all functions have an inverse. For example, the function f(x) = x^2 does not have an inverse, because it is not one-to-one.

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

References

[1] "Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Functions in Real-World Applications" by Wolfram MathWorld

Glossary

Domain: The set of all possible inputs for a function.
Range: The set of all possible outputs for a function.
Function: A relation between a set of inputs and a set of possible outputs, where each input is assigned to a unique output.
Relation: A set of ordered pairs, where each pair consists of an input and an output.
One-to-one: A function where each output corresponds to exactly one input.
Onto: A function where each output corresponds to at least one input.
Bijection: A function that is both one-to-one and onto.
Inverse: A function that undoes the original function.