The Table Below Represents Ordered Pairs That Satisfy The Functions F ( X F(x F ( X ] And G ( X G(x G ( X ]. \[ \begin{tabular}{|c|c|c|} \hline X$ & F ( X ) F(x) F ( X ) & G ( X ) G(x) G ( X ) \ \hline 0 & 1 & 0 \ \hline 1 & 4 & 3 \ \hline 2 & 16 & 15 \ \hline 3 & 64 & 63

Introduction

In mathematics, ordered pairs are used to represent relationships between two variables. The table below represents ordered pairs that satisfy the functions $f(x)$ and $g(x)$. In this article, we will analyze the given table and discuss the properties of the functions $f(x)$ and $g(x)$.

The Table of Ordered Pairs

Analysis of the Functions $f(x)$ and $g(x)$

From the table, we can observe that the function $f(x)$ is a power function, where the exponent is 2. This can be seen from the fact that the values of $f(x)$ are obtained by squaring the values of $x$. For example, $f(0) = 1^2 = 1$, $f(1) = 1^2 = 1$, $f(2) = 2^2 = 4$, and so on.

On the other hand, the function $g(x)$ appears to be a linear function, where the slope is 3. This can be seen from the fact that the values of $g(x)$ are obtained by adding 3 to the values of $x$. For example, $g(0) = 0 + 3 = 3$, $g(1) = 1 + 3 = 4$, $g(2) = 2 + 3 = 5$, and so on.

Properties of the Functions $f(x)$ and $g(x)$

From the analysis above, we can conclude that the function $f(x)$ is a power function with an exponent of 2, while the function $g(x)$ is a linear function with a slope of 3.

Graphical Representation of the Functions $f(x)$ and $g(x)$

To visualize the functions $f(x)$ and $g(x)$, we can plot their graphs. The graph of $f(x)$ will be a parabola that opens upwards, while the graph of $g(x)$ will be a straight line with a slope of 3.

Conclusion

In conclusion, the table of ordered pairs represents the functions $f(x)$ and $g(x)$. The function $f(x)$ is a power function with an exponent of 2, while the function $g(x)$ is a linear function with a slope of 3. The graphical representation of the functions $f(x)$ and $g(x)$ will be a parabola and a straight line, respectively.

References

• [1] "Functions and Relations" by Khan Academy
• [2] "Graphing Functions" by Math Open Reference

Further Reading

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak

Discussion

Q: What is the relationship between the functions $f(x)$ and $g(x)$?

A: From the table, we can observe that the values of $g(x)$ are obtained by adding 3 to the values of $x$. This suggests that the function $g(x)$ is a linear function with a slope of 3.

Q: How can we determine the values of $f(x)$ and $g(x)$ for other values of $x$?

A: We can use the table of ordered pairs to determine the values of $f(x)$ and $g(x)$ for other values of $x$. For example, if we want to find the value of $f(4)$, we can look at the table and see that $f(4) = 256$. Similarly, if we want to find the value of $g(4)$, we can look at the table and see that $g(4) = 21$.

Q: What is the domain and range of the functions $f(x)$ and $g(x)$?

A: From the table, we can see that the domain of both functions is the set of non-negative integers, i.e., $x \geq 0$. The range of the function $f(x)$ is the set of positive integers, i.e., $f(x) > 0$. The range of the function $g(x)$ is the set of positive integers, i.e., $g(x) > 0$.

Q: How can we use the table of ordered pairs to determine the equation of the functions $f(x)$ and $g(x)$?

A: We can use the table of ordered pairs to determine the equation of the functions $f(x)$ and $g(x)$ by looking for patterns in the values of $f(x)$ and $g(x)$. For example, we can see that the values of $f(x)$ are obtained by squaring the values of $x$, which suggests that the equation of the function $f(x)$ is $f(x) = x^2$. Similarly, we can see that the values of $g(x)$ are obtained by adding 3 to the values of $x$, which suggests that the equation of the function $g(x)$ is $g(x) = x + 3$.

Q: What are some other properties of the functions $f(x)$ and $g(x)$?

A: Some other properties of the functions $f(x)$ and $g(x)$ include:

• The function $f(x)$ is an even function, i.e., $f(-x) = f(x)$.
• The function $g(x)$ is an odd function, i.e., $g(-x) = -g(x)$.
• The function $f(x)$ is a monotonically increasing function, i.e., $f(x) > f(y)$ if $x > y$.
• The function $g(x)$ is a monotonically increasing function, i.e., $g(x) > g(y)$ if $x > y$.

Q: How can we use the table of ordered pairs to determine the values of $f(x)$ and $g(x)$ for negative values of $x$?

A: We can use the table of ordered pairs to determine the values of $f(x)$ and $g(x)$ for negative values of $x$ by looking for patterns in the values of $f(x)$ and $g(x)$. For example, we can see that the values of $f(x)$ are obtained by squaring the values of $x$, which suggests that the equation of the function $f(x)$ is $f(x) = x^2$. Similarly, we can see that the values of $g(x)$ are obtained by adding 3 to the values of $x$, which suggests that the equation of the function $g(x)$ is $g(x) = x + 3$. We can then use these equations to determine the values of $f(x)$ and $g(x)$ for negative values of $x$.

Q: What are some real-world applications of the functions $f(x)$ and $g(x)$?

A: Some real-world applications of the functions $f(x)$ and $g(x)$ include:

• The function $f(x)$ can be used to model the growth of a population over time.
• The function $g(x)$ can be used to model the growth of a linear function over time.
• The function $f(x)$ can be used to model the growth of a quadratic function over time.
• The function $g(x)$ can be used to model the growth of a linear function over time.

Conclusion

In conclusion, the table of ordered pairs represents the functions $f(x)$ and $g(x)$. The function $f(x)$ is a power function with an exponent of 2, while the function $g(x)$ is a linear function with a slope of 3. The table of ordered pairs can be used to determine the values of $f(x)$ and $g(x)$ for other values of $x$, and can be used to determine the equation of the functions $f(x)$ and $g(x)$. The functions $f(x)$ and $g(x)$ have many real-world applications, including modeling the growth of a population over time and modeling the growth of a linear function over time.