Consider The Function Represented By The Table.${ \begin{tabular}{|c|c|} \hline X X X & Π ( X ) \pi(x) Π ( X ) \ \hline 2 & 6 \ \hline 7 & 3 \ \hline 9 & 5 \ \hline \end{tabular} }$The Ordered Pair Given In The Bottom Row Can Be Written Using Function

Mar 12, 2025 by ADMIN 253 views

**Understanding the Function Represented by the Table**

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

The Table

The table given represents a function with two inputs, $x$ and $\pi(x)$ , where $\pi(x)$ is the number of prime numbers less than or equal to $x$ . The table has three rows, each representing a different input and its corresponding output.

$x$	$\pi(x)$
2	6
7	3
9	5

Analyzing the Table

From the table, we can see that the function has three ordered pairs: $(2, 6), (7, 3),$ and $(9, 5)$ . These pairs represent the input and output values of the function.

For $x = 2$ , the output is $\pi(2) = 6$ .
For $x = 7$ , the output is $\pi(7) = 3$ .
For $x = 9$ , the output is $\pi(9) = 5$ .

Properties of the Function

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

Domain: The domain of the function is the set of all possible input values, which in this case is $\{2, 7, 9\}$ .
Range: The range of the function is the set of all possible output values, which in this case is $\{3, 5, 6\}$ .
One-to-One: The function is one-to-one, meaning that each input corresponds to exactly one output.
Onto: The function is not onto, meaning that not every possible output value is achieved.

Graphing the Function

To visualize the function, we can plot the ordered pairs on a coordinate plane. The graph of the function will be a set of points, each representing an input and output value.

Conclusion

In conclusion, the function represented by the table is a one-to-one function with a domain of $\{2, 7, 9\}$ and a range of $\{3, 5, 6\}$ . The function is not onto, meaning that not every possible output value is achieved. Understanding the properties of a function is essential in mathematics, as it helps us to analyze and interpret the relationships between variables.

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, quadratic, polynomial, and rational functions, to name a few. Each type of function has its own properties and characteristics, and understanding these properties is essential in mathematics and its applications.

Real-World Applications

Functions are used in many real-world applications, including:

Science: Functions are used to model the behavior of physical systems, such as the motion of objects and the flow of fluids.
Engineering: Functions are used to design and optimize systems, such as bridges and buildings.
Economics: Functions are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Final Thoughts

Introduction

In our previous article, we explored the function represented by a given table and understood its properties. In this article, we will answer some frequently asked questions about the function and its properties.

Q: What is the domain of the function?

A: The domain of the function is the set of all possible input values, which in this case is $\{2, 7, 9\}$ .

Q: What is the range of the function?

A: The range of the function is the set of all possible output values, which in this case is $\{3, 5, 6\}$ .

Q: Is the function one-to-one?

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

Q: Is the function onto?

A: No, the function is not onto, meaning that not every possible output value is achieved.

Q: How can we graph the function?

A: We can graph the function by plotting the ordered pairs on a coordinate plane. The graph of the function will be a set of points, each representing an input and output value.

Q: What are some real-world applications of functions?

A: Functions are used in many real-world applications, including:

Science: Functions are used to model the behavior of physical systems, such as the motion of objects and the flow of fluids.
Engineering: Functions are used to design and optimize systems, such as bridges and buildings.
Economics: Functions are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Q: What are some common types of functions?

A: Some common types of functions include:

Linear functions: These functions have a constant rate of change and can be represented by a linear equation.
Quadratic functions: These functions have a parabolic shape and can be represented by a quadratic equation.
Polynomial functions: These functions have a general form of $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ and can be represented by a polynomial equation.
Rational functions: These functions have a general form of $f(x) = \frac{p(x)}{q(x)}$ and can be represented by a rational equation.

Q: How can we determine if a function is one-to-one or onto?

A: To determine if a function is one-to-one or onto, we can use the following criteria:

One-to-one: A function is one-to-one if each input corresponds to exactly one output.
Onto: A function is onto if every possible output value is achieved.

Q: What are some common mistakes to avoid when working with functions?

A: Some common mistakes to avoid when working with functions include:

Confusing the domain and range: Make sure to distinguish between the domain and range of a function.
Not checking for one-to-one or onto properties: Make sure to check if a function is one-to-one or onto before using it in a calculation.
Not using the correct notation: Make sure to use the correct notation when representing a function, such as $f(x)$ or $y = f(x)$ .

Conclusion

In conclusion, understanding the function represented by the table is essential in mathematics and its applications. By answering some frequently asked questions about the function and its properties, we can gain a deeper understanding of the subject and avoid common mistakes.