Which Of The Following Lists Of Ordered Pairs Is A Function?A. { (2,5), (3,6), (6,9)$}$B. { (2,5), (3,6), (2,1)$}$C. { (1,2), (4,0), (3,5), (4,3)$}$D. { (-1,2), (2,3), (3,1), (2,5)$}$
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 describing a relationship between two sets of values. A function is often represented as a set of ordered pairs, where each ordered pair consists of an input value and its corresponding output value. In this article, we will explore which of the given lists of ordered pairs represents 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 describing a relationship between two sets of values. A function is often represented as a set of ordered pairs, where each ordered pair consists of an input value and its corresponding output value. For a relation to be a function, each input value must correspond to only one output value.
Properties of a Function
There are several properties of a function that we need to consider when determining whether a relation is a function. These properties include:
- Domain: The domain of a function is the set of all possible input values.
- Range: The range of a function is the set of all possible output values.
- One-to-One: A function is one-to-one if each input value corresponds to only one output value.
- Onto: A function is onto if each output value corresponds to at least one input value.
Analyzing the Given Lists of Ordered Pairs
Now that we have a good understanding of what a function is and its properties, let's analyze the given lists of ordered pairs to determine which one represents a function.
A. {(2,5), (3,6), (6,9)$}$
This list of ordered pairs represents a function because each input value corresponds to only one output value. The domain of this function is {2, 3, 6} and the range is {5, 6, 9}. This function is one-to-one and onto.
B. {(2,5), (3,6), (2,1)$}$
This list of ordered pairs does not represent a function because the input value 2 corresponds to two different output values, 5 and 1. This violates the one-to-one property of a function.
C. {(1,2), (4,0), (3,5), (4,3)$}$
This list of ordered pairs represents a function because each input value corresponds to only one output value. The domain of this function is {1, 4, 3} and the range is {2, 0, 5, 3}. However, this function is not one-to-one because the input value 4 corresponds to two different output values, 0 and 3.
D. {(-1,2), (2,3), (3,1), (2,5)$}$
This list of ordered pairs does not represent a function because the input value 2 corresponds to two different output values, 3 and 5. This violates the one-to-one property of a function.
Conclusion
In conclusion, only list A represents a function because each input value corresponds to only one output value. The other lists of ordered pairs do not represent functions because they violate the one-to-one property of a function.
Key Takeaways
- 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 often represented as a set of ordered pairs, where each ordered pair consists of an input value and its corresponding output value.
- For a relation to be a function, each input value must correspond to only one output value.
- The domain and range of a function are the set of all possible input and output values, respectively.
- A function is one-to-one if each input value corresponds to only one output value.
- A function is onto if each output value corresponds to at least one input value.
Final Thoughts
In the previous article, we explored the concept of a function and its properties. We also analyzed the given lists of ordered pairs to determine which one represents a function. In this article, we will answer some frequently asked questions (FAQs) about functions.
Q: What is a function?
A: 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.
Q: What are the properties of a function?
A: The properties of a function include:
- Domain: The domain of a function is the set of all possible input values.
- Range: The range of a function is the set of all possible output values.
- One-to-One: A function is one-to-one if each input value corresponds to only one output value.
- Onto: A function is onto if each output value corresponds to at least one input value.
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 value corresponds to only one output value. If each input value corresponds to only one output value, then the relation is a function.
Q: What is the difference between a function and a relation?
A: A function is a relation that satisfies the one-to-one property, meaning that each input value corresponds to only one output value. A relation, on the other hand, is a set of ordered pairs that do not necessarily satisfy the one-to-one property.
Q: Can a function have multiple output values for the same input value?
A: No, a function cannot have multiple output values for the same input value. If a function has multiple output values for the same input value, then it is not a function.
Q: Can a function have an empty domain or range?
A: Yes, a function can have an empty domain or range. However, if the domain or range is empty, then the function is not defined.
Q: Can a function be represented as a table or a graph?
A: Yes, a function can be represented as a table or a graph. A table or graph can be used to visualize the relationship between the input and output values of a function.
Q: How do I find the domain and range of a function?
A: To find the domain and range of a function, you need to examine the set of ordered pairs that make up the function. The domain is the set of all possible input values, and the range is the set of all possible output values.
Q: Can a function be one-to-one and onto at the same time?
A: Yes, a function can be one-to-one and onto at the same time. In fact, a function that is both one-to-one and onto is called a bijective function.
Q: Can a function be represented as a formula or an equation?
A: Yes, a function can be represented as a formula or an equation. A formula or equation can be used to describe the relationship between the input and output values of a function.
Conclusion
In conclusion, functions are an important concept in mathematics that describe a relationship between a set of inputs and a set of possible outputs. We have answered some frequently asked questions (FAQs) about functions, including what a function is, the properties of a function, and how to determine if a relation is a function. We hope that this article has provided a clear understanding of functions and their properties.
Key Takeaways
- 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 often represented as a set of ordered pairs, where each ordered pair consists of an input value and its corresponding output value.
- For a relation to be a function, each input value must correspond to only one output value.
- The domain and range of a function are the set of all possible input and output values, respectively.
- A function is one-to-one if each input value corresponds to only one output value.
- A function is onto if each output value corresponds to at least one input value.
Final Thoughts
In this article, we have answered some frequently asked questions (FAQs) about functions. We hope that this article has provided a clear understanding of functions and their properties. If you have any further questions or need clarification on any of the concepts discussed in this article, please don't hesitate to ask.