Select The Correct Answer.Exponential Function { F $}$ Is Represented By The Table Below.${ \begin{tabular}{|l|c|c|c|c|c|} \hline X X X & -1 & 0 & 1 & 2 & 3 \ \hline F ( X ) F(x) F ( X ) & 78 & 24 & 6 & 0 & -2 \ \hline \end{tabular} }$Function
Understanding Exponential Functions
Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as x and f(x). These functions are characterized by their ability to grow or decay at an exponential rate, often represented by the equation f(x) = ab^x, where a and b are constants. In this article, we will explore the concept of exponential functions and how to select the correct answer based on a given table.
Representing Exponential Functions with Tables
Exponential functions can be represented using tables, where the input values (x) are listed in one column and the corresponding output values (f(x)) are listed in another column. The table below represents an exponential function f(x) with input values ranging from -1 to 3.
| x | f(x) |
|---|---|
| -1 | 78 |
| 0 | 24 |
| 1 | 6 |
| 2 | 0 |
| 3 | -2 |
Analyzing the Table
To select the correct answer, we need to analyze the table and identify the pattern of the exponential function. Looking at the table, we can see that the output values (f(x)) are decreasing as the input values (x) increase. This suggests that the exponential function is a decreasing function.
Identifying the Exponential Function
To identify the exponential function, we need to examine the relationship between the input values (x) and the output values (f(x)). We can see that the output values (f(x)) are decreasing by a factor of 4 as the input values (x) increase by 1. This suggests that the exponential function is of the form f(x) = a(4)^(-x), where a is a constant.
Finding the Value of a
To find the value of a, we can use the fact that f(0) = 24. Substituting x = 0 into the equation f(x) = a(4)^(-x), we get:
f(0) = a(4)^(-0) f(0) = a(1) f(0) = a Since f(0) = 24, we can conclude that a = 24.
Writing the Exponential Function
Now that we have found the value of a, we can write the exponential function as:
f(x) = 24(4)^(-x)
Conclusion
In this article, we have explored the concept of exponential functions and how to select the correct answer based on a given table. We have analyzed the table, identified the pattern of the exponential function, and written the exponential function in the form f(x) = a(4)^(-x). We have also found the value of a using the fact that f(0) = 24.
Key Takeaways
- Exponential functions are a type of mathematical function that describes a relationship between two variables.
- Exponential functions can be represented using tables, where the input values (x) are listed in one column and the corresponding output values (f(x)) are listed in another column.
- To select the correct answer, we need to analyze the table and identify the pattern of the exponential function.
- The exponential function can be written in the form f(x) = a(4)^(-x), where a is a constant.
Common Mistakes to Avoid
- Not analyzing the table and identifying the pattern of the exponential function.
- Not using the fact that f(0) = 24 to find the value of a.
- Not writing the exponential function in the correct form.
Real-World Applications
Exponential functions have many real-world applications, including:
- Modeling population growth and decay.
- Modeling chemical reactions.
- Modeling financial investments.
Practice Problems
- Find the value of a in the exponential function f(x) = a(2)^(-x), given that f(0) = 12.
- Write the exponential function f(x) = a(3)^(-x) in the form f(x) = a(1/3)^x.
- Find the value of x in the exponential function f(x) = 24(4)^(-x), given that f(x) = 6.
Solutions
- f(0) = a(2)^(-0) f(0) = a(1) f(0) = a Since f(0) = 12, we can conclude that a = 12.
- f(x) = a(3)^(-x) f(x) = a(1/3)^x
- f(x) = 24(4)^(-x) f(x) = 24(1/4)^x Since f(x) = 6, we can conclude that 24(1/4)^x = 6. Simplifying the equation, we get: (1/4)^x = 1/4 x = 1
Conclusion
Frequently Asked Questions
Q: What is an exponential function?
A: An exponential function is a type of mathematical function that describes a relationship between two variables, typically denoted as x and f(x). These functions are characterized by their ability to grow or decay at an exponential rate, often represented by the equation f(x) = ab^x, where a and b are constants.
Q: How do I represent an exponential function with a table?
A: To represent an exponential function with a table, you need to list the input values (x) in one column and the corresponding output values (f(x)) in another column. The table below represents an exponential function f(x) with input values ranging from -1 to 3.
| x | f(x) |
|---|---|
| -1 | 78 |
| 0 | 24 |
| 1 | 6 |
| 2 | 0 |
| 3 | -2 |
Q: How do I identify the exponential function from a table?
A: To identify the exponential function from a table, you need to examine the relationship between the input values (x) and the output values (f(x)). You can see that the output values (f(x)) are decreasing by a factor of 4 as the input values (x) increase by 1. This suggests that the exponential function is of the form f(x) = a(4)^(-x), where a is a constant.
Q: How do I find the value of a in an exponential function?
A: To find the value of a in an exponential function, you can use the fact that f(0) = a. Substituting x = 0 into the equation f(x) = a(4)^(-x), you get:
f(0) = a(4)^(-0) f(0) = a(1) f(0) = a
Since f(0) = 24, you can conclude that a = 24.
Q: How do I write an exponential function in the correct form?
A: To write an exponential function in the correct form, you need to identify the base (b) and the exponent (x). The base is the constant that is raised to the power of the exponent. The exponent is the variable that is multiplied by the base. For example, the exponential function f(x) = 24(4)^(-x) has a base of 4 and an exponent of -x.
Q: What are some real-world applications of exponential functions?
A: Exponential functions have many real-world applications, including:
- Modeling population growth and decay.
- Modeling chemical reactions.
- Modeling financial investments.
Q: How do I solve exponential function problems?
A: To solve exponential function problems, you need to follow these steps:
- Identify the base and the exponent.
- Use the fact that f(0) = a to find the value of a.
- Write the exponential function in the correct form.
- Use the properties of exponents to simplify the function.
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 analyzing the table and identifying the pattern of the exponential function.
- Not using the fact that f(0) = a to find the value of a.
- Not writing the exponential function in the correct form.
Q: How do I practice solving exponential function problems?
A: To practice solving exponential function problems, you can try the following:
- Use online resources, such as Khan Academy or Mathway, to practice solving exponential function problems.
- Work with a tutor or a study group to practice solving exponential function problems.
- Use real-world examples, such as population growth or chemical reactions, to practice solving exponential function problems.
Q: What are some advanced topics in exponential functions?
A: Some advanced topics in exponential functions include:
- Logarithmic functions.
- Exponential decay.
- Exponential growth.
Q: How do I apply exponential functions to real-world problems?
A: To apply exponential functions to real-world problems, you need to follow these steps:
- Identify the problem and the variables involved.
- Use the properties of exponents to model the problem.
- Use the exponential function to solve the problem.
- Interpret the results and draw conclusions.
Conclusion
In this article, we have explored the concept of exponential functions and how to select the correct answer based on a given table. We have also provided a Q&A section to help readers understand the concept better. We have covered topics such as identifying the exponential function, finding the value of a, writing the exponential function in the correct form, and solving exponential function problems. We have also provided practice problems and solutions to help readers understand the concept better.