The Table Shows Three Unique, Discrete Functions. \[ \begin{tabular}{|c|c|c|c|} \hline X$ & F ( X ) F(x) F ( X ) & G ( X ) G(x) G ( X ) & H ( X ) H(x) H ( X ) \ \hline -1 & 2 & 2 & 2 \ \hline 0 & 3 & 3 & 3 \ \hline 3 & 2 & 0 & 4 \ \hline 8 & -3 & -25 & 5 \ \hline 15 & -10 & -204 & 6

Introduction

In mathematics, functions are a fundamental concept that plays a crucial role in various branches of study, including algebra, calculus, and analysis. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. In this article, we will analyze three unique, discrete functions, denoted as $f(x)$, $g(x)$, and $h(x)$, and presented in a table format.

The Table of Functions

$x$	$f(x)$	$g(x)$	$h(x)$
-1	2	2	2
0	3	3	3
3	2	0	4
8	-3	-25	5
15	-10	-204	6

Analyzing the Functions

Function $f(x)$

The function $f(x)$ is a discrete function that takes an input $x$ and returns an output $f(x)$. From the table, we can observe that the function $f(x)$ has a range of values from 2 to -10. The function appears to be decreasing as the input $x$ increases.

Function $g(x)$

The function $g(x)$ is another discrete function that takes an input $x$ and returns an output $g(x)$. From the table, we can observe that the function $g(x)$ has a range of values from 2 to -204. The function appears to be decreasing rapidly as the input $x$ increases.

Function $h(x)$

The function $h(x)$ is a discrete function that takes an input $x$ and returns an output $h(x)$. From the table, we can observe that the function $h(x)$ has a range of values from 2 to 6. The function appears to be increasing as the input $x$ increases.

Comparing the Functions

Comparison of $f(x)$ and $g(x)$

A comparison of the functions $f(x)$ and $g(x)$ reveals that both functions have a similar range of values, but the function $g(x)$ has a much larger range of values. Additionally, the function $g(x)$ appears to be decreasing more rapidly than the function $f(x)$.

Comparison of $f(x)$ and $h(x)$

A comparison of the functions $f(x)$ and $h(x)$ reveals that the function $f(x)$ has a smaller range of values than the function $h(x)$. Additionally, the function $h(x)$ appears to be increasing as the input $x$ increases, whereas the function $f(x)$ appears to be decreasing.

Comparison of $g(x)$ and $h(x)$

A comparison of the functions $g(x)$ and $h(x)$ reveals that the function $g(x)$ has a much larger range of values than the function $h(x)$. Additionally, the function $g(x)$ appears to be decreasing rapidly as the input $x$ increases, whereas the function $h(x)$ appears to be increasing.

Conclusion

In conclusion, the three unique, discrete functions, $f(x)$, $g(x)$, and $h(x)$, presented in the table, have distinct characteristics. The function $f(x)$ has a smaller range of values and appears to be decreasing as the input $x$ increases. The function $g(x)$ has a much larger range of values and appears to be decreasing rapidly as the input $x$ increases. The function $h(x)$ has a moderate range of values and appears to be increasing as the input $x$ increases.

Future Work

Further analysis of the functions $f(x)$, $g(x)$, and $h(x)$ could involve:

• Determining the domain of each function: The domain of a function is the set of all possible input values. Determining the domain of each function could provide valuable insights into the behavior of each function.
• Analyzing the behavior of each function: Analyzing the behavior of each function could involve determining the rate of change of each function, identifying any patterns or trends, and determining any limitations or restrictions on the behavior of each function.
• Comparing the functions to other functions: Comparing the functions $f(x)$, $g(x)$, and $h(x)$ to other functions could provide valuable insights into the behavior of each function and could help to identify any relationships or patterns between the functions.

References

• [1] Calculus: A First Course, by Michael Spivak
• [2] Discrete Mathematics, by Kenneth H. Rosen
• [3] Functions, by Wolfram MathWorld

Appendix

The following is a list of the input values and corresponding output values for each function:

$x$	$f(x)$	$g(x)$	$h(x)$
-1	2	2	2
0	3	3	3
3	2	0	4
8	-3	-25	5
15	-10	-204	6

Q: What are the three unique, discrete functions presented in the table?

A: The three unique, discrete functions presented in the table are $f(x)$, $g(x)$, and $h(x)$.

Q: What is the range of values for each function?

A: The range of values for each function is as follows:

• $f(x)$: 2 to -10
• $g(x)$: 2 to -204
• $h(x)$: 2 to 6

Q: How do the functions $f(x)$ and $g(x)$ compare?

A: The functions $f(x)$ and $g(x)$ have a similar range of values, but the function $g(x)$ has a much larger range of values. Additionally, the function $g(x)$ appears to be decreasing more rapidly than the function $f(x)$.

Q: How do the functions $f(x)$ and $h(x)$ compare?

A: The functions $f(x)$ and $h(x)$ have different ranges of values. The function $f(x)$ has a smaller range of values than the function $h(x)$. Additionally, the function $h(x)$ appears to be increasing as the input $x$ increases, whereas the function $f(x)$ appears to be decreasing.

Q: How do the functions $g(x)$ and $h(x)$ compare?

A: The functions $g(x)$ and $h(x)$ have different ranges of values. The function $g(x)$ has a much larger range of values than the function $h(x)$. Additionally, the function $g(x)$ appears to be decreasing rapidly as the input $x$ increases, whereas the function $h(x)$ appears to be increasing.

Q: What is the domain of each function?

A: The domain of each function is not explicitly stated in the table. However, based on the input values presented, the domain of each function appears to be the set of integers from -1 to 15.

Q: What is the behavior of each function?

A: The behavior of each function is as follows:

• $f(x)$: The function $f(x)$ appears to be decreasing as the input $x$ increases.
• $g(x)$: The function $g(x)$ appears to be decreasing rapidly as the input $x$ increases.
• $h(x)$: The function $h(x)$ appears to be increasing as the input $x$ increases.

Q: What are some possible applications of these functions?

A: Some possible applications of these functions include:

• Modeling real-world phenomena: The functions $f(x)$, $g(x)$, and $h(x)$ could be used to model real-world phenomena, such as population growth, chemical reactions, or electrical circuits.
• Solving optimization problems: The functions $f(x)$, $g(x)$, and $h(x)$ could be used to solve optimization problems, such as finding the maximum or minimum value of a function.
• Analyzing data: The functions $f(x)$, $g(x)$, and $h(x)$ could be used to analyze data, such as identifying trends or patterns in a dataset.

Q: What are some possible extensions of this work?

A: Some possible extensions of this work include:

• Determining the domain of each function: Determining the domain of each function could provide valuable insights into the behavior of each function.
• Analyzing the behavior of each function: Analyzing the behavior of each function could involve determining the rate of change of each function, identifying any patterns or trends, and determining any limitations or restrictions on the behavior of each function.
• Comparing the functions to other functions: Comparing the functions $f(x)$, $g(x)$, and $h(x)$ to other functions could provide valuable insights into the behavior of each function and could help to identify any relationships or patterns between the functions.

Q: What are some possible future directions for research?

A: Some possible future directions for research include:

• Developing new functions: Developing new functions that exhibit interesting or useful properties could lead to new insights and applications.
• Analyzing the behavior of complex systems: Analyzing the behavior of complex systems, such as those involving multiple functions or variables, could lead to a deeper understanding of the underlying dynamics.
• Developing new mathematical tools: Developing new mathematical tools, such as new functions or algorithms, could lead to new insights and applications in a variety of fields.