The Function \[$ F \$\] Is Defined By The Following Rule:$\[ F(x) = \left(\frac{1}{9}\right)^x \\]Find \[$ F(x) \$\] For Each \[$ X \$\]-value In The Table.$\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -2
Introduction to the Function f(x)
The function f(x) is defined by the rule f(x) = (1/9)^x. This function is an exponential function, where the base is 1/9 and the exponent is x. The function f(x) is used to model various real-world phenomena, such as population growth, chemical reactions, and financial transactions.
Understanding Exponential Functions
Exponential functions are a type of mathematical function that describes a relationship between two variables, where one variable is raised to a power of the other variable. In the case of the function f(x) = (1/9)^x, the base is 1/9 and the exponent is x. This means that as x increases, the value of f(x) decreases, and as x decreases, the value of f(x) increases.
Calculating f(x) for Each x-Value in the Table
To find f(x) for each x-value in the table, we need to substitute each x-value into the function f(x) = (1/9)^x.
x = -2
To find f(-2), we substitute x = -2 into the function f(x) = (1/9)^x.
f(-2) = (1/9)^(-2) = 9^2 = 81
x = -1
To find f(-1), we substitute x = -1 into the function f(x) = (1/9)^x.
f(-1) = (1/9)^(-1) = 9 = 9
x = 0
To find f(0), we substitute x = 0 into the function f(x) = (1/9)^x.
f(0) = (1/9)^0 = 1
x = 1
To find f(1), we substitute x = 1 into the function f(x) = (1/9)^x.
f(1) = (1/9)^1 = 1/9
x = 2
To find f(2), we substitute x = 2 into the function f(x) = (1/9)^x.
f(2) = (1/9)^2 = 1/81
Conclusion
In conclusion, the function f(x) = (1/9)^x is an exponential function that describes a relationship between two variables, where one variable is raised to a power of the other variable. We have calculated f(x) for each x-value in the table, and the results are as follows:
| x | f(x) |
|---|---|
| -2 | 81 |
| -1 | 9 |
| 0 | 1 |
| 1 | 1/9 |
| 2 | 1/81 |
The function f(x) has many applications in mathematics, including modeling population growth, chemical reactions, and financial transactions. It is an important concept in mathematics and has many real-world implications.
Real-World Applications of the Function f(x)
The function f(x) = (1/9)^x has many real-world applications, including:
- Population Growth: The function f(x) can be used to model population growth, where the base is the growth rate and the exponent is the time period.
- Chemical Reactions: The function f(x) can be used to model chemical reactions, where the base is the rate of reaction and the exponent is the time period.
- Financial Transactions: The function f(x) can be used to model financial transactions, where the base is the interest rate and the exponent is the time period.
Limitations of the Function f(x)
The function f(x) = (1/9)^x has some limitations, including:
- Domain: The domain of the function f(x) is all real numbers, but the function is only defined for x ≥ 0.
- Range: The range of the function f(x) is all positive real numbers.
- Asymptotes: The function f(x) has a horizontal asymptote at y = 0.
Conclusion
In conclusion, the function f(x) = (1/9)^x is an exponential function that describes a relationship between two variables, where one variable is raised to a power of the other variable. We have calculated f(x) for each x-value in the table, and the results are as follows:
| x | f(x) |
|---|---|
| -2 | 81 |
| -1 | 9 |
| 0 | 1 |
| 1 | 1/9 |
| 2 | 1/81 |
The function f(x) has many applications in mathematics, including modeling population growth, chemical reactions, and financial transactions. It is an important concept in mathematics and has many real-world implications.
Q: What is the function f(x) defined by?
A: The function f(x) is defined by the rule f(x) = (1/9)^x.
Q: What type of function is f(x)?
A: The function f(x) is an exponential function.
Q: What is the base of the function f(x)?
A: The base of the function f(x) is 1/9.
Q: What is the exponent of the function f(x)?
A: The exponent of the function f(x) is x.
Q: What is the domain of the function f(x)?
A: The domain of the function f(x) is all real numbers.
Q: What is the range of the function f(x)?
A: The range of the function f(x) is all positive real numbers.
Q: What is the horizontal asymptote of the function f(x)?
A: The horizontal asymptote of the function f(x) is y = 0.
Q: How do I calculate f(x) for a given x-value?
A: To calculate f(x) for a given x-value, substitute the x-value into the function f(x) = (1/9)^x.
Q: What are some real-world applications of the function f(x)?
A: Some real-world applications of the function f(x) include modeling population growth, chemical reactions, and financial transactions.
Q: What are some limitations of the function f(x)?
A: Some limitations of the function f(x) include its domain, range, and asymptotes.
Q: Can I use the function f(x) to model any type of growth or decay?
A: Yes, the function f(x) can be used to model any type of growth or decay, as long as the base and exponent are defined.
Q: How do I graph the function f(x)?
A: To graph the function f(x), use a graphing calculator or software to plot the function for various x-values.
Q: Can I use the function f(x) to solve any type of problem?
A: Yes, the function f(x) can be used to solve a wide range of problems, including those involving exponential growth and decay.
Q: What are some common mistakes to avoid when working with the function f(x)?
A: Some common mistakes to avoid when working with the function f(x) include:
- Incorrectly calculating f(x) for a given x-value
- Using the wrong base or exponent
- Not considering the domain and range of the function
- Not graphing the function correctly
Q: How do I determine if a given function is an exponential function?
A: To determine if a given function is an exponential function, check if it can be written in the form f(x) = a^x, where a is a positive real number.
Q: Can I use the function f(x) to model any type of periodic behavior?
A: No, the function f(x) is not suitable for modeling periodic behavior, as it is an exponential function.
Q: How do I use the function f(x) to model population growth?
A: To use the function f(x) to model population growth, substitute the growth rate into the function f(x) = (1/9)^x, where x is the time period.
Q: Can I use the function f(x) to model any type of financial transaction?
A: Yes, the function f(x) can be used to model any type of financial transaction, including interest rates and investment returns.
Q: How do I use the function f(x) to model chemical reactions?
A: To use the function f(x) to model chemical reactions, substitute the rate of reaction into the function f(x) = (1/9)^x, where x is the time period.
Q: Can I use the function f(x) to model any type of physical system?
A: Yes, the function f(x) can be used to model any type of physical system, including those involving exponential growth and decay.
Q: How do I determine if a given function is an exponential function with a base of 1/9?
A: To determine if a given function is an exponential function with a base of 1/9, check if it can be written in the form f(x) = (1/9)^x.