Write An Exponential Function Represented By The Table.$\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 2 & 3 & 4 & 5 & 6 \\ \hline $g(x)$ & 4.5 & 13.5 & 40.5 & 121.5 & 364.5 \\ \hline \end{tabular} \\]

Introduction

In mathematics, an exponential function is a function that has the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. These functions are used to model real-world phenomena, such as population growth, chemical reactions, and financial transactions. In this article, we will explore how to represent an exponential function using a table and discuss the characteristics of exponential functions.

Understanding the Table

The table provided represents an exponential function $g(x)$ with input values of $x$ ranging from 2 to 6. The corresponding output values of $g(x)$ are 4.5, 13.5, 40.5, 121.5, and 364.5, respectively. To determine the exponential function represented by this table, we need to identify the base and the exponent.

Identifying the Base and Exponent

Upon examining the table, we notice that each output value is obtained by multiplying the previous output value by a constant factor. This suggests that the function is exponential in nature. To determine the base and exponent, we can use the following steps:

1. Identify the common ratio: The common ratio is the constant factor by which each output value is multiplied to obtain the next output value. In this case, the common ratio is 3, as each output value is 3 times the previous output value.
2. Determine the base: The base of an exponential function is the number that is raised to the power of the exponent. In this case, the base is 3, as each output value is obtained by raising 3 to a power.
3. Determine the exponent: The exponent is the power to which the base is raised. In this case, the exponent is $x-2$, as each output value is obtained by raising 3 to the power of $x-2$.

Representing the Exponential Function

Using the base and exponent determined above, we can represent the exponential function as:

$$g(x) = 3^{x-2}$$

This function represents the exponential function $g(x)$ with input values of $x$ ranging from 2 to 6.

Characteristics of Exponential Functions

Exponential functions have several characteristics that are important to understand:

• Growth rate: Exponential functions grow at an increasing rate, meaning that the output values increase rapidly as the input values increase.
• Asymptotes: Exponential functions have asymptotes, which are horizontal 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.

Real-World Applications of Exponential Functions

Exponential functions have numerous real-world applications, including:

• Population growth: Exponential functions can be used to model population growth, where the population increases at an increasing rate.
• Chemical reactions: Exponential functions can be used to model chemical reactions, where the concentration of a substance increases at an increasing rate.
• Financial transactions: Exponential functions can be used to model financial transactions, such as compound interest, where the interest rate increases at an increasing rate.

Conclusion

In conclusion, exponential functions are an important concept in mathematics that have numerous real-world applications. By understanding how to represent an exponential function using a table and identifying the base and exponent, we can model real-world phenomena and make predictions about future outcomes. The characteristics of exponential functions, such as growth rate, asymptotes, and domain and range, are also important to understand. By applying exponential functions to real-world problems, we can gain a deeper understanding of the world around us.

Example Problems

Problem 1

Represent the exponential function $f(x) = 2^x$ using a table.

Solution

To represent the exponential function $f(x) = 2^x$ using a table, we can use the following steps:

1. Identify the base: The base of the function is 2.
2. Determine the exponent: The exponent is $x$, as each output value is obtained by raising 2 to the power of $x$.
3. Create a table: Create a table with input values of $x$ ranging from 0 to 5 and corresponding output values of $f(x)$.

The resulting table is:

$x$	$f(x)$
0	1
1	2
2	4
3	8
4	16
5	32

Problem 2

Determine the base and exponent of the exponential function $g(x) = 5^{x-1}$.

Solution

To determine the base and exponent of the exponential function $g(x) = 5^{x-1}$, we can use the following steps:

1. Identify the base: The base of the function is 5.
2. Determine the exponent: The exponent is $x-1$, as each output value is obtained by raising 5 to the power of $x-1$.

Practice Problems

Problem 1

Represent the exponential function $f(x) = 3^x$ using a table.

Problem 2

Determine the base and exponent of the exponential function $g(x) = 2^{x+1}$.

Problem 3

Solve the equation $2^x = 16$.

Problem 4

Solve the equation $3^x = 27$.

Problem 5

Represent the exponential function $f(x) = 4^x$ using a table.

Problem 6

Determine the base and exponent of the exponential function $g(x) = 6^{x-2}$.

Problem 7

Solve the equation $5^x = 125$.

Problem 8

Solve the equation $2^x = 32$.

Problem 9

Represent the exponential function $f(x) = 7^x$ using a table.

Problem 10

Determine the base and exponent of the exponential function $g(x) = 8^{x+2}$.

Glossary

• Exponential function: A function that has the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable.
• Base: The number that is raised to the power of the exponent in an exponential function.
• Exponent: The power to which the base is raised in an exponential function.
• Common ratio: The constant factor by which each output value is multiplied to obtain the next output value in an exponential function.
• Asymptote: A horizontal line that an exponential function approaches but never touches.
• Domain: The set of all input values for which an exponential function is defined.
• Range: The set of all output values for which an exponential function is defined.
  

Introduction

Exponential functions are a fundamental concept in mathematics that have numerous real-world applications. However, they can be challenging to understand and work with, especially for those who are new to the subject. In this article, we will address some of the most frequently asked questions about exponential functions, providing clear and concise answers to help you better understand this important mathematical concept.

Q: What is an exponential function?

A: An exponential function is a function that has the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. Exponential functions are used to model real-world phenomena, such as population growth, chemical reactions, and financial transactions.

Q: How do I represent an exponential function using a table?

A: To represent an exponential function using a table, you need to identify the base and exponent. The base is the number that is raised to the power of the exponent, while the exponent is the power to which the base is raised. You can use the following steps to create a table:

1. Identify the base: The base of the function is the number that is raised to the power of the exponent.
2. Determine the exponent: The exponent is the power to which the base is raised.
3. Create a table: Create a table with input values of $x$ ranging from 0 to 5 and corresponding output values of $f(x)$.

Q: What is the difference between an exponential function and a linear function?

A: An exponential function and a linear function are two different types of functions that have distinct characteristics. A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. An exponential function, on the other hand, has the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$. You can use the following steps:

1. Use logarithms: Use logarithms to rewrite the equation in a form that is easier to solve.
2. Simplify the equation: Simplify the equation by combining like terms.
3. Solve for $x$: Solve for $x$ by isolating the variable.

Q: What is the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, while the range is all positive real numbers. This means that an exponential function can take on any positive value, but it cannot take on a negative value.

Q: How do I graph an exponential function?

A: To graph an exponential function, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph. The graph of an exponential function will have a characteristic "S" shape, with the function increasing rapidly as the input values increase.

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

A: Exponential functions have numerous real-world applications, including:

• Population growth: Exponential functions can be used to model population growth, where the population increases at an increasing rate.
• Chemical reactions: Exponential functions can be used to model chemical reactions, where the concentration of a substance increases at an increasing rate.
• Financial transactions: Exponential functions can be used to model financial transactions, such as compound interest, where the interest rate increases at an increasing rate.

Q: How do I determine the base and exponent of an exponential function?

A: To determine the base and exponent of an exponential function, you need to examine the function and identify the base and exponent. The base is the number that is raised to the power of the exponent, while the exponent is the power to which the base is raised.

Q: What is the difference between an exponential function and a power function?

A: An exponential function and a power function are two different types of functions that have distinct characteristics. A power function has the form $f(x) = x^n$, where $n$ is a positive integer. An exponential function, on the other hand, has the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable.

Q: How do I use exponential functions to model real-world phenomena?

A: To use exponential functions to model real-world phenomena, you need to identify the base and exponent of the function. You can then use the function to make predictions about future outcomes.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

• Not identifying the base and exponent: Failing to identify the base and exponent of an exponential function can lead to incorrect solutions.
• Not using logarithms: Failing to use logarithms to solve exponential equations can lead to incorrect solutions.
• Not checking the domain and range: Failing to check the domain and range of an exponential function can lead to incorrect solutions.

Conclusion

In conclusion, exponential functions are a fundamental concept in mathematics that have numerous real-world applications. By understanding how to represent an exponential function using a table, identifying the base and exponent, and solving exponential equations, you can model real-world phenomena and make predictions about future outcomes. Remember to avoid common mistakes, such as not identifying the base and exponent, not using logarithms, and not checking the domain and range.

Glossary

• Exponential function: A function that has the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable.
• Base: The number that is raised to the power of the exponent in an exponential function.
• Exponent: The power to which the base is raised in an exponential function.
• Common ratio: The constant factor by which each output value is multiplied to obtain the next output value in an exponential function.
• Asymptote: A horizontal line that an exponential function approaches but never touches.
• Domain: The set of all input values for which an exponential function is defined.
• Range: The set of all output values for which an exponential function is defined.

Practice Problems

Problem 1

Represent the exponential function $f(x) = 3^x$ using a table.

Problem 2

Determine the base and exponent of the exponential function $g(x) = 2^{x+1}$.

Problem 3

Solve the equation $2^x = 16$.

Problem 4

Solve the equation $3^x = 27$.

Problem 5

Represent the exponential function $f(x) = 4^x$ using a table.

Problem 6

Determine the base and exponent of the exponential function $g(x) = 6^{x-2}$.

Problem 7

Solve the equation $5^x = 125$.

Problem 8

Solve the equation $2^x = 32$.

Problem 9

Represent the exponential function $f(x) = 7^x$ using a table.

Problem 10

Determine the base and exponent of the exponential function $g(x) = 8^{x+2}$.

Additional Resources

• Exponential function calculator: A calculator that can be used to calculate exponential functions.
• Exponential function grapher: A grapher that can be used to graph exponential functions.
• Exponential function solver: A solver that can be used to solve exponential equations.

Conclusion

In conclusion, exponential functions are a fundamental concept in mathematics that have numerous real-world applications. By understanding how to represent an exponential function using a table, identifying the base and exponent, and solving exponential equations, you can model real-world phenomena and make predictions about future outcomes. Remember to avoid common mistakes, such as not identifying the base and exponent, not using logarithms, and not checking the domain and range.