Determining the Correct Statement

Oct 13, 2025 by magicalceylontours.com 41 views

Introduction

In mathematics, functions are used to represent relationships between variables. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function is typically represented by a mathematical expression or a table. In this article, we will analyze a given function represented by a table and determine which statement correctly describes its behavior.

Function Representation

The function $g(x)$ is represented by the table below:

$x$	$-1$	$0$	$1$	$2$	$3$
$g(x)$	$15$	$3$	$0$	$-\frac{3}{4}$	$-\frac{15}{16}$

Analyzing the Function

To analyze the function, we need to examine its behavior at each point in the domain. From the table, we can see that the function has the following values:

$g(-1) = 15$
$g(0) = 3$
$g(1) = 0$
$g(2) = -\frac{3}{4}$
$g(3) = -\frac{15}{16}$

Determining the Correct Statement

Based on the values of the function, we can determine which statement correctly describes its behavior. The possible statements are:

The function is increasing on the interval $[-1, 0]$ .
The function is decreasing on the interval $[0, 1]$ .
The function is increasing on the interval $[1, 2]$ .
The function is decreasing on the interval $[2, 3]$ .
The function is a constant function.

Statement Analysis

To determine which statement is correct, we need to examine the behavior of the function on each interval.

Interval $[-1, 0]$ : The function is increasing on this interval because $g(-1) = 15 > g(0) = 3$ .
Interval $[0, 1]$ : The function is decreasing on this interval because $g(0) = 3 > g(1) = 0$ .
Interval $[1, 2]$ : The function is decreasing on this interval because $g(1) = 0 > g(2) = -\frac{3}{4}$ .
Interval $[2, 3]$ : The function is decreasing on this interval because $g(2) = -\frac{3}{4} > g(3) = -\frac{15}{16}$ .

Conclusion

Based on the analysis, we can conclude that the function is decreasing on the intervals $[0, 1]$ , $[1, 2]$ , and $[2, 3]$ . Therefore, the correct statement is:

The function is decreasing on the interval $[0, 1]$ .
The function is decreasing on the interval $[1, 2]$ .
The function is decreasing on the interval $[2, 3]$ .

Final Answer

Introduction

In our previous article, we analyzed a function represented by a table and determined which statement correctly describes its behavior. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

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 typically represented by a mathematical expression or a table.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible y-values.

Q: How do we determine the behavior of a function?

A: We can determine the behavior of a function by examining its values at different points in the domain. We can also use mathematical expressions or graphs to visualize the function's behavior.

Q: What is an increasing function?

A: An increasing function is a function that takes on larger values as the input increases. In other words, if x increases, then f(x) also increases.

Q: What is a decreasing function?

A: A decreasing function is a function that takes on smaller values as the input increases. In other words, if x increases, then f(x) also decreases.

Q: How do we determine if a function is increasing or decreasing?

A: We can determine if a function is increasing or decreasing by examining its values at different points in the domain. We can also use mathematical expressions or graphs to visualize the function's behavior.

Q: What is a constant function?

A: A constant function is a function that takes on the same value for all input values. In other words, f(x) = c for all x.

Q: How do we determine if a function is a constant function?

A: We can determine if a function is a constant function by examining its values at different points in the domain. If the function takes on the same value for all input values, then it is a constant function.

Q: What is the significance of analyzing a function's behavior?

A: Analyzing a function's behavior is important because it helps us understand how the function changes as the input changes. This can be useful in a variety of applications, such as modeling real-world phenomena or optimizing functions.

Conclusion

In this Q&A article, we have provided additional insights and clarification on the concepts of function representation and analysis. We hope that this article has been helpful in understanding the behavior of functions and how to analyze them.

Final Answer

The final answer is that analyzing a function's behavior is an important step in understanding how the function changes as the input changes. By examining the function's values at different points in the domain, we can determine its behavior and use this information to make informed decisions.