Which Table Represents An Exponential Function Of The Form Y = B X Y=b^x Y = B X When 0 \textless B \textless 1 0\ \textless \ B\ \textless \ 1 0 \textless B \textless 1 ? \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline -3 & 1 27 \frac{1}{27} 27 1 ​ \ \hline -2 & 1 9 \frac{1}{9} 9 1 ​ \ \hline -1 &

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields such as science, engineering, and economics. In this article, we will explore the concept of exponential functions of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$. We will also examine the characteristics of these functions and determine which table represents an exponential function of this form.

What is an Exponential Function?

An exponential function is a mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y=b^x$, where $b$ is a positive constant and $x$ is the variable. The value of $y$ is determined by raising the base $b$ to the power of $x$.

Characteristics of Exponential Functions

Exponential functions have several key characteristics that distinguish them from other types of functions. Some of the most important characteristics of exponential functions include:

• Growth or Decay: Exponential functions can exhibit either growth or decay, depending on the value of the base $b$. If $b>1$, the function exhibits growth, while if $0\ \textless \ b\ \text< 1$, the function exhibits decay.
• Asymptotes: Exponential functions can have asymptotes, which are horizontal or vertical lines that the function approaches but never touches.
• Domain and Range: The domain of an exponential function is all real numbers, while the range is all positive real numbers.

Exponential Functions of the Form $y=b^x$

In this article, we are interested in exponential functions of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$. These functions exhibit decay, meaning that as $x$ increases, $y$ decreases. The base $b$ determines the rate of decay, with smaller values of $b$ resulting in faster decay.

Which Table Represents an Exponential Function?

We are given three tables to consider, each representing a different relationship between $x$ and $y$. We need to determine which table represents an exponential function of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$.

Table 1

$x$	$y$
-3	1/27
-2	1/9
-1	1/3
0	1
1	3
2	9
3	27

Table 2

$x$	$y$
-3	27
-2	9
-1	3
0	1
1	1/3
2	1/9
3	1/27

Table 3

$x$	$y$
-3	1/27
-2	1/9
-1	1/3
0	1
1	3
2	9
3	27

Analyzing the Tables

To determine which table represents an exponential function of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$, we need to examine the relationships between $x$ and $y$ in each table.

• Table 1: In this table, we can see that as $x$ increases, $y$ increases exponentially. The base $b$ appears to be 3, since $3^x$ is the relationship between $x$ and $y$.
• Table 2: In this table, we can see that as $x$ increases, $y$ decreases exponentially. The base $b$ appears to be 3, since $3^x$ is the relationship between $x$ and $y$.
• Table 3: In this table, we can see that as $x$ increases, $y$ increases exponentially. The base $b$ appears to be 1/3, since $(1/3)^x$ is the relationship between $x$ and $y$.

Conclusion

Based on our analysis, we can conclude that Table 3 represents an exponential function of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$. The base $b$ is 1/3, and the relationship between $x$ and $y$ is given by $(1/3)^x$.

Discussion

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields such as science, engineering, and economics. In this article, we have explored the concept of exponential functions of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$. We have also examined the characteristics of these functions and determined which table represents an exponential function of this form.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Exponential Functions" by Khan Academy
• [3] "Exponential Functions" by Wolfram MathWorld

Further Reading

• "Exponential Functions" by MIT OpenCourseWare
• "Exponential Functions" by University of California, Berkeley
• "Exponential Functions" by Stanford University

In our previous article, we explored the concept of exponential functions of the form $y=b^x$, where $0\ \textless \ b\ \text< 1$. We also examined the characteristics of these functions and determined which table represents an exponential function of this form. In this article, we will answer some frequently asked questions about exponential functions.

Q: What is an exponential function?

A: An exponential function is a mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y=b^x$, where $b$ is a positive constant and $x$ is the variable.

Q: What are the characteristics of exponential functions?

A: Exponential functions have several key characteristics that distinguish them from other types of functions. Some of the most important characteristics of exponential functions include:

• Growth or Decay: Exponential functions can exhibit either growth or decay, depending on the value of the base $b$. If $b>1$, the function exhibits growth, while if $0\ \textless \ b\ \text< 1$, the function exhibits decay.
• Asymptotes: Exponential functions can have asymptotes, which are horizontal or vertical lines that the function approaches but never touches.
• Domain and Range: The domain of an exponential function is all real numbers, while the range is all positive real numbers.

Q: How do I determine if a function is exponential?

A: To determine if a function is exponential, you need to examine the relationship between the input and output values. If the output values increase or decrease exponentially as the input values increase or decrease, then the function is likely exponential.

Q: What is the base of an exponential function?

A: The base of an exponential function is the constant $b$ in the equation $y=b^x$. The base determines the rate of growth or decay of the function.

Q: Can the base of an exponential function be negative?

A: No, the base of an exponential function cannot be negative. The base must be a positive constant, since the function is defined as $y=b^x$.

Q: Can the base of an exponential function be 1?

A: Yes, the base of an exponential function can be 1. In this case, the function is a constant function, since $1^x=1$ for all values of $x$.

Q: Can the base of an exponential function be greater than 1?

A: Yes, the base of an exponential function can be greater than 1. In this case, the function exhibits growth, since the output values increase exponentially as the input values increase.

Q: Can the base of an exponential function be less than 1?

A: Yes, the base of an exponential function can be less than 1. In this case, the function exhibits decay, since the output values decrease exponentially as the input values increase.

Q: How do I graph an exponential function?

A: To graph an exponential function, you need to plot the points $(x, y)$ for various values of $x$. You can use a graphing calculator or a computer program to graph the function.

Q: Can I use exponential functions in real-world applications?

A: Yes, exponential functions have many real-world applications. Some examples include:

• Population growth: Exponential functions can be used to model population growth, since the population can grow exponentially over time.
• Financial applications: Exponential functions can be used to model financial applications, such as compound interest and depreciation.
• Science and engineering: Exponential functions can be used to model scientific and engineering applications, such as radioactive decay and electrical circuits.

Conclusion

Exponential functions are a fundamental concept in mathematics, and they have many real-world applications. In this article, we have answered some frequently asked questions about exponential functions, including their characteristics, how to determine if a function is exponential, and how to graph an exponential function. We hope this article has been helpful in understanding exponential functions.