Select The Correct Answer.Which Of These Tables Represents A Function? \[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline X$ & Y Y Y & X X X & Y Y Y & X X X & Y Y Y & X X X & Y Y Y \ \hline -1 & 15 & 0 & 4 & 4 & 0 & 6 & -2 \ \hline -2 & 6 & 6 & 15 & -1 & -2 & 4 & 0

Mar 12, 2025 by ADMIN 260 views

**Understanding Functions in Mathematics**

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 which of the given tables represents a function.

What is a Function?

A function is a relation between a set of inputs (domain) and a set of possible outputs (range). It is a way of describing a relationship between two variables, where each input corresponds to exactly one output. In other words, for every input, there is only one corresponding output.

Key Characteristics of a Function

To determine if a relation represents a function, we need to check if it satisfies the following key characteristics:

Each input corresponds to exactly one output: For every input, there must be only one corresponding output.
No two different inputs have the same output: If two different inputs have the same output, then the relation is not a function.

Analyzing the Tables

Let's analyze the given tables to determine which one represents a function.

Table 1

$x$	$y$	$x$	$y$	$x$	$y$	$x$	$y$
-1	15	0	4	4	0	6	-2
-2	6	6	15	-1	-2	4	0

Is Table 1 a Function?

Let's check if Table 1 satisfies the key characteristics of a function.

Each input corresponds to exactly one output: In Table 1, we can see that each input corresponds to only one output. For example, the input -1 corresponds to the output 15, and the input 4 corresponds to the output 0.
No two different inputs have the same output: In Table 1, we can see that no two different inputs have the same output. For example, the input -1 has a different output (15) than the input 4 (0).

Based on these observations, we can conclude that Table 1 represents a function.

Table 2

$x$	$y$	$x$	$y$	$x$	$y$	$x$	$y$
-1	15	0	4	4	0	6	-2
-2	6	6	15	-1	-2	4	0

Is Table 2 a Function?

Let's check if Table 2 satisfies the key characteristics of a function.

Each input corresponds to exactly one output: In Table 2, we can see that each input corresponds to only one output. For example, the input -1 corresponds to the output 15, and the input 4 corresponds to the output 0.
No two different inputs have the same output: In Table 2, we can see that no two different inputs have the same output. For example, the input -1 has a different output (15) than the input 4 (0).

Based on these observations, we can conclude that Table 2 represents a function.

Table 3

$x$	$y$	$x$	$y$	$x$	$y$	$x$	$y$
-1	15	0	4	4	0	6	-2
-2	6	6	15	-1	-2	4	0

Is Table 3 a Function?

Let's check if Table 3 satisfies the key characteristics of a function.

Each input corresponds to exactly one output: In Table 3, we can see that each input corresponds to only one output. For example, the input -1 corresponds to the output 15, and the input 4 corresponds to the output 0.
No two different inputs have the same output: In Table 3, we can see that no two different inputs have the same output. For example, the input -1 has a different output (15) than the input 4 (0).

Based on these observations, we can conclude that Table 3 represents a function.

Conclusion

In conclusion, all three tables represent functions. Each input corresponds to exactly one output, and no two different inputs have the same output. Therefore, we can say that all three tables satisfy the key characteristics of a function.

Why is it Important to Understand Functions?

Understanding functions is crucial in mathematics and other fields. Functions are used to describe relationships between variables, and they are essential in modeling real-world phenomena. In mathematics, functions are used to solve equations, optimize problems, and analyze data. In other fields, functions are used to model population growth, economic systems, and physical systems.

Real-World Applications of Functions

Functions have numerous real-world applications. Some examples include:

Population growth: Functions are used to model population growth and decline.
Economic systems: Functions are used to model economic systems and predict economic outcomes.
Physical systems: Functions are used to model physical systems and predict their behavior.
Computer science: Functions are used in computer science to write efficient and effective code.

Conclusion

In this article, we will answer some frequently asked questions 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 variables, where each input corresponds to exactly one output.

Q: What are the key characteristics of a function?

A: The key characteristics of a function are:

Each input corresponds to exactly one output: For every input, there must be only one corresponding output.
No two different inputs have the same output: If two different inputs have the same output, then the relation is not a function.

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

A: To determine if a relation represents a function, you need to check if it satisfies the key characteristics of a function. You can do this by:

Checking if each input corresponds to exactly one output: For every input, check if there is only one corresponding output.
Checking if no two different inputs have the same output: Check if any two different inputs have the same output.

Q: What are some examples of functions?

A: Some examples of functions include:

Linear functions: These are functions that can be represented by a linear equation, such as f(x) = 2x + 3.
Quadratic functions: These are functions that can be represented by a quadratic equation, such as f(x) = x^2 + 2x + 1.
Polynomial functions: These are functions that can be represented by a polynomial equation, such as f(x) = x^3 + 2x^2 + x + 1.

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

A: Some real-world applications of functions include:

Population growth: Functions are used to model population growth and decline.
Economic systems: Functions are used to model economic systems and predict economic outcomes.
Physical systems: Functions are used to model physical systems and predict their behavior.
Computer science: Functions are used in computer science to write efficient and effective code.

Q: How do I graph a function?

A: To graph a function, you need to:

Determine the domain and range: Determine the set of inputs (domain) and the set of possible outputs (range).
Plot the points: Plot the points on a coordinate plane, using the x-axis for the input and the y-axis for the output.
Draw the graph: Draw the graph of the function, using the plotted points as a guide.

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

A: A function is a relation between a set of inputs (domain) and a set of possible outputs (range), where each input corresponds to exactly one output. A relation is a set of ordered pairs, where each pair represents a relationship between two variables.

Q: Can a relation be a function?

A: Yes, a relation can be a function if it satisfies the key characteristics of a function. In other words, if each input corresponds to exactly one output, and no two different inputs have the same output, then the relation is a function.

Q: Can a function be a relation?

A: Yes, a function is a type of relation. In other words, a function is a relation that satisfies the key characteristics of a function.

Conclusion

In conclusion, functions are an essential concept in mathematics and other fields. Understanding functions is crucial in modeling real-world phenomena and solving problems. In this article, we answered some frequently asked questions about functions, including what a function is, how to determine if a relation represents a function, and some real-world applications of functions.