Understanding the Ordered Pairs

Oct 13, 2025 by windailys.com 55 views

Introduction

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 variables, where each input corresponds to exactly one output. In this article, we will explore the function represented by the given table and understand its properties.

The Table

The table given represents a function with four ordered pairs:

$x$	$f(x)$
2	6
7	3
9	5

Understanding the Ordered Pairs

Each ordered pair in the table represents a point on the graph of the function. The first element of the pair, $x$ , is the input, and the second element, $f(x)$ , is the output. For example, the first ordered pair $(2, 6)$ means that when the input is $2$ , the output is $6$ .

Function Notation

The function represented by the table can be written in function notation as $f(x)$ . This notation means that the function $f$ takes an input $x$ and returns an output $f(x)$ .

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. From the table, we can see that the domain of the function is $\{2, 7, 9\}$ , and the range is $\{3, 5, 6\}$ .

Properties of the Function

From the table, we can observe the following properties of the function:

One-to-One: The function is one-to-one, meaning that each input corresponds to exactly one output. For example, when $x = 2$ , the output is $6$ , and when $x = 7$ , the output is $3$ .
Onto: The function is onto, meaning that every output value in the range is obtained for some input value in the domain. For example, the output $6$ is obtained when $x = 2$ , and the output $3$ is obtained when $x = 7$ .
Not Injective: The function is not injective, meaning that different input values can correspond to the same output value. For example, when $x = 7$ , the output is $3$ , and when $x = 9$ , the output is also $5$ .

Graph of the Function

The graph of the function is a visual representation of the function's behavior. It is a set of points on the coordinate plane, where each point represents an ordered pair in the table.

Conclusion

In this article, we have explored the function represented by the given table. We have understood the properties of the function, including its domain and range, and have observed its behavior on the graph. The function is one-to-one and onto, but not injective. We have also seen how the function can be written in function notation and how it can be represented graphically.

Further Discussion

The function represented by the table is a simple example of a mathematical function. However, there are many other types of functions, including linear functions, quadratic functions, and polynomial functions. Each type of function has its own properties and behavior, and understanding these properties is essential for working with functions in mathematics.

References

[1] "Functions" by Khan Academy
[2] "Functions" by Math Open Reference
[3] "Functions" by Wolfram MathWorld

Additional Resources

[1] "Functions" by MIT OpenCourseWare
[2] "Functions" by Stanford University
[3] "Functions" by University of California, Berkeley

Glossary

Domain: The set of all possible input values of a function.
Range: The set of all possible output values of a function.
One-to-One: A function is one-to-one if each input corresponds to exactly one output.
Onto: A function is onto if every output value in the range is obtained for some input value in the domain.
Injective: A function is injective if different input values correspond to different output values.
Function Notation: A way of writing a function using the notation $f(x)$ , where $f$ is the function and $x$ is the input.
Frequently Asked Questions (FAQs) about the Function Represented by the Table ====================================================================================

Q: What is the domain of the function represented by the table?

A: The domain of the function represented by the table is the set of all possible input values, which is $\{2, 7, 9\}$ .

Q: What is the range of the function represented by the table?

A: The range of the function represented by the table is the set of all possible output values, which is $\{3, 5, 6\}$ .

Q: Is the function represented by the table one-to-one?

A: Yes, the function represented by the table is one-to-one, meaning that each input corresponds to exactly one output.

Q: Is the function represented by the table onto?

A: Yes, the function represented by the table is onto, meaning that every output value in the range is obtained for some input value in the domain.

Q: Is the function represented by the table injective?

A: No, the function represented by the table is not injective, meaning that different input values can correspond to the same output value.

Q: How can we represent the function represented by the table graphically?

A: The function represented by the table can be represented graphically as a set of points on the coordinate plane, where each point represents an ordered pair in the table.

Q: What is the significance of the function represented by the table in real-world applications?

A: The function represented by the table can be used to model real-world situations, such as the relationship between the number of items and their cost, or the relationship between the temperature and the amount of ice cream sold.

Q: Can we extend the function represented by the table to include more input values?

A: Yes, we can extend the function represented by the table to include more input values by adding more ordered pairs to the table.

Q: How can we use the function represented by the table to solve problems?

A: We can use the function represented by the table to solve problems by using it to find the output value for a given input value, or by using it to find the input value for a given output value.

Q: What are some common mistakes to avoid when working with the function represented by the table?

A: Some common mistakes to avoid when working with the function represented by the table include:

Assuming that the function is one-to-one or onto without checking the domain and range.
Failing to check the domain and range of the function before using it to solve problems.
Using the function to solve problems without considering the context of the problem.

Q: How can we use technology to help us work with the function represented by the table?

A: We can use technology, such as graphing calculators or computer software, to help us work with the function represented by the table by:

Graphing the function to visualize its behavior.
Using the function to solve problems and check our answers.
Extending the function to include more input values.

Q: What are some real-world applications of the function represented by the table?

A: Some real-world applications of the function represented by the table include:

Modeling the relationship between the number of items and their cost.
Modeling the relationship between the temperature and the amount of ice cream sold.
Modeling the relationship between the number of customers and the amount of revenue generated.

Q: How can we use the function represented by the table to make predictions?

A: We can use the function represented by the table to make predictions by using it to find the output value for a given input value, or by using it to find the input value for a given output value.

Q: What are some limitations of the function represented by the table?

A: Some limitations of the function represented by the table include:

The function is only defined for the input values in the domain.
The function is only defined for the output values in the range.
The function may not be accurate for all input values.

Q: How can we use the function represented by the table to solve optimization problems?

A: We can use the function represented by the table to solve optimization problems by using it to find the maximum or minimum value of the function.

Q: What are some common optimization problems that can be solved using the function represented by the table?

A: Some common optimization problems that can be solved using the function represented by the table include:

Finding the maximum or minimum value of the function.
Finding the input value that corresponds to a given output value.
Finding the output value that corresponds to a given input value.

Q: How can we use the function represented by the table to solve systems of equations?

A: We can use the function represented by the table to solve systems of equations by using it to find the solution to the system.

Q: What are some common systems of equations that can be solved using the function represented by the table?

A: Some common systems of equations that can be solved using the function represented by the table include:

Systems of linear equations.
Systems of quadratic equations.
Systems of polynomial equations.

Q: How can we use the function represented by the table to solve differential equations?

A: We can use the function represented by the table to solve differential equations by using it to find the solution to the differential equation.

Q: What are some common differential equations that can be solved using the function represented by the table?

A: Some common differential equations that can be solved using the function represented by the table include:

First-order linear differential equations.
Second-order linear differential equations.
Nonlinear differential equations.