Given The Function: $f(x)=\left(\frac{1}{8}\right)^x$What Is The Value Of $f(x$\] When $x$ Is A Specific Number?

The given function, $f(x)=\left(\frac{1}{8}\right)^x$, is an exponential function with a base of $\frac{1}{8}$. This type of function is characterized by its rapid growth or decay, depending on the value of the exponent. In this case, the function is decreasing as $x$ increases, since the base is less than 1.

Key Characteristics of Exponential Functions

Exponential functions have several key characteristics that are essential to understand when working with them. These include:

• Rapid growth or decay: Exponential functions can grow or decay rapidly, depending on the value of the base and the exponent.
• Asymptotic behavior: Exponential functions can have asymptotic behavior, where the function approaches a certain value as the input increases or decreases without bound.
• Sensitivity to the base: The base of an exponential function can greatly affect its behavior, with small changes in the base leading to significant changes in the function's growth or decay rate.

Analyzing the Given Function

The given function, $f(x)=\left(\frac{1}{8}\right)^x$, is a specific type of exponential function with a base of $\frac{1}{8}$. To understand its behavior, we can analyze the function's key characteristics.

• Rapid decay: Since the base is less than 1, the function decays rapidly as $x$ increases.
• Asymptotic behavior: As $x$ approaches negative infinity, the function approaches 1, since $\left(\frac{1}{8}\right)^{-\infty}$ approaches 1.
• Sensitivity to the base: The base of $\frac{1}{8}$ is relatively small, which means the function decays rapidly.

Finding the Value of $f(x)$ for a Specific Number

To find the value of $f(x)$ for a specific number, we need to substitute the value of $x$ into the function. For example, if we want to find the value of $f(3)$, we would substitute $x=3$ into the function:

$f(3)=\left(\frac{1}{8}\right)^3$

To evaluate this expression, we can use the fact that $\left(\frac{1}{8}\right)^3=\frac{1}{8^3}=\frac{1}{512}$.

Example Calculations

To illustrate the behavior of the function, let's calculate the value of $f(x)$ for a few specific values of $x$:

• $f(0)=\left(\frac{1}{8}\right)^0=1$
• $f(1)=\left(\frac{1}{8}\right)^1=\frac{1}{8}$
• $f(2)=\left(\frac{1}{8}\right)^2=\frac{1}{64}$
• $f(3)=\left(\frac{1}{8}\right)^3=\frac{1}{512}$

As we can see, the function decays rapidly as $x$ increases, with the value of $f(x)$ approaching 0 as $x$ approaches infinity.

Conclusion

In conclusion, the given function, $f(x)=\left(\frac{1}{8}\right)^x$, is an exponential function with a base of $\frac{1}{8}$. The function decays rapidly as $x$ increases, with the value of $f(x)$ approaching 0 as $x$ approaches infinity. To find the value of $f(x)$ for a specific number, we can substitute the value of $x$ into the function and evaluate the resulting expression.

Key Takeaways

• Exponential functions can have rapid growth or decay, depending on the value of the base and the exponent.
• The base of an exponential function can greatly affect its behavior, with small changes in the base leading to significant changes in the function's growth or decay rate.
• To find the value of an exponential function for a specific number, we can substitute the value of $x$ into the function and evaluate the resulting expression.

Further Exploration

For further exploration, we can consider the following questions:

• What happens to the function as $x$ approaches positive infinity?
• How does the function behave for negative values of $x$?
• Can we find a general formula for the value of $f(x)$ for any value of $x$?

In this article, we will continue to explore the function $f(x)=\left(\frac{1}{8}\right)^x$ and answer some common questions about its behavior and applications.

Q: What is the domain of the function $f(x)=\left(\frac{1}{8}\right)^x$?

A: The domain of the function $f(x)=\left(\frac{1}{8}\right)^x$ is all real numbers, since the base $\frac{1}{8}$ is a positive number and the exponent $x$ can take on any real value.

Q: What is the range of the function $f(x)=\left(\frac{1}{8}\right)^x$?

A: The range of the function $f(x)=\left(\frac{1}{8}\right)^x$ is all positive real numbers, since the base $\frac{1}{8}$ is a positive number and the exponent $x$ can take on any real value.

Q: How does the function $f(x)=\left(\frac{1}{8}\right)^x$ behave as $x$ approaches positive infinity?

A: As $x$ approaches positive infinity, the function $f(x)=\left(\frac{1}{8}\right)^x$ approaches 0, since the base $\frac{1}{8}$ is less than 1.

Q: How does the function $f(x)=\left(\frac{1}{8}\right)^x$ behave as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the function $f(x)=\left(\frac{1}{8}\right)^x$ approaches infinity, since the base $\frac{1}{8}$ is less than 1.

Q: Can we find a general formula for the value of $f(x)$ for any value of $x$?

A: Yes, we can find a general formula for the value of $f(x)$ for any value of $x$. The formula is:

$f(x)=\left(\frac{1}{8}\right)^x$

This formula is valid for all real values of $x$.

Q: How does the function $f(x)=\left(\frac{1}{8}\right)^x$ relate to other exponential functions?

A: The function $f(x)=\left(\frac{1}{8}\right)^x$ is a specific type of exponential function with a base of $\frac{1}{8}$. Other exponential functions have different bases, such as $e^x$ or $2^x$. The behavior of these functions can be compared and contrasted with the behavior of the function $f(x)=\left(\frac{1}{8}\right)^x$.

Q: What are some common applications of the function $f(x)=\left(\frac{1}{8}\right)^x$?

A: The function $f(x)=\left(\frac{1}{8}\right)^x$ has several common applications in mathematics and other fields, including:

• Modeling population growth: The function $f(x)=\left(\frac{1}{8}\right)^x$ can be used to model population growth in a population that is declining rapidly.
• Modeling chemical reactions: The function $f(x)=\left(\frac{1}{8}\right)^x$ can be used to model the rate of a chemical reaction that is decreasing rapidly.
• Modeling financial data: The function $f(x)=\left(\frac{1}{8}\right)^x$ can be used to model financial data that is decreasing rapidly.

Conclusion

In conclusion, the function $f(x)=\left(\frac{1}{8}\right)^x$ is a specific type of exponential function with a base of $\frac{1}{8}$. The function decays rapidly as $x$ increases, with the value of $f(x)$ approaching 0 as $x$ approaches infinity. The function has several common applications in mathematics and other fields, including modeling population growth, modeling chemical reactions, and modeling financial data.

Key Takeaways

• Exponential functions can have rapid growth or decay, depending on the value of the base and the exponent.
• The base of an exponential function can greatly affect its behavior, with small changes in the base leading to significant changes in the function's growth or decay rate.
• To find the value of an exponential function for a specific number, we can substitute the value of $x$ into the function and evaluate the resulting expression.

Further Exploration

For further exploration, we can consider the following questions:

• What happens to the function as $x$ approaches positive infinity?
• How does the function behave for negative values of $x$?
• Can we find a general formula for the value of $f(x)$ for any value of $x$?

By exploring these questions and others, we can gain a deeper understanding of the behavior of exponential functions and their applications in mathematics and other fields.