Which Is An Exponential Decay Function?A. $f(x)=\frac{3}{4}\left(\frac{7}{4}\right)^x$B. $f(x)=\frac{2}{3}\left(\frac{4}{5}\right)^{-x}$C. $f(x)=\frac{3}{2}\left(\frac{8}{7}\right)^{-x}$D.

What is an Exponential Decay Function?

An exponential decay function is a mathematical function that describes a relationship between two variables, where one variable decreases exponentially as the other variable increases. In other words, the function models a situation where a quantity decreases at a rate proportional to its current value. Exponential decay functions are commonly used in various fields, including physics, engineering, economics, and biology, to model phenomena such as population growth, radioactive decay, and chemical reactions.

Characteristics of Exponential Decay Functions

Exponential decay functions have several key characteristics that distinguish them from other types of functions. These characteristics include:

• Exponential form: Exponential decay functions are typically written in the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable.
• Decaying rate: The rate at which the function decays is determined by the base $b$. If $b$ is between 0 and 1, the function decays exponentially. If $b$ is greater than 1, the function grows exponentially.
• Initial value: The initial value of the function is determined by the constant $a$. This is the value of the function when $x$ is equal to 0.

Examples of Exponential Decay Functions

Here are some examples of exponential decay functions:

• $f(x) = 2(0.5)^x$
• $f(x) = 3(0.8)^{-x}$
• $f(x) = 4(0.9)^x$

Which of the Following Functions is an Exponential Decay Function?

Now that we have a good understanding of what an exponential decay function is and what its characteristics are, let's take a look at the options provided:

A. $f(x)=\frac{3}{4}\left(\frac{7}{4}\right)^x$ B. $f(x)=\frac{2}{3}\left(\frac{4}{5}\right)^{-x}$ C. $f(x)=\frac{3}{2}\left(\frac{8}{7}\right)^{-x}$ D. $f(x)=\frac{3}{2}\left(\frac{7}{8}\right)^x$

To determine which of these functions is an exponential decay function, we need to examine each option carefully.

Option A: $f(x)=\frac{3}{4}\left(\frac{7}{4}\right)^x$

This function is in the form $f(x) = ab^x$, where $a = \frac{3}{4}$ and $b = \frac{7}{4}$. Since $b$ is greater than 1, this function grows exponentially, not decays. Therefore, this is not an exponential decay function.

Option B: $f(x)=\frac{2}{3}\left(\frac{4}{5}\right)^{-x}$

This function is also in the form $f(x) = ab^x$, where $a = \frac{2}{3}$ and $b = \frac{4}{5}$. Since $b$ is between 0 and 1, this function decays exponentially. Therefore, this is an exponential decay function.

Option C: $f(x)=\frac{3}{2}\left(\frac{8}{7}\right)^{-x}$

This function is in the form $f(x) = ab^x$, where $a = \frac{3}{2}$ and $b = \frac{8}{7}$. Since $b$ is greater than 1, this function grows exponentially, not decays. Therefore, this is not an exponential decay function.

Option D: $f(x)=\frac{3}{2}\left(\frac{7}{8}\right)^x$

This function is in the form $f(x) = ab^x$, where $a = \frac{3}{2}$ and $b = \frac{7}{8}$. Since $b$ is between 0 and 1, this function decays exponentially. Therefore, this is an exponential decay function.

Conclusion

In conclusion, the exponential decay function among the options provided is:

• Option B: $f(x)=\frac{2}{3}\left(\frac{4}{5}\right)^{-x}$
• Option D: $f(x)=\frac{3}{2}\left(\frac{7}{8}\right)^x$

Q: What is an exponential decay function?

A: An exponential decay function is a mathematical function that describes a relationship between two variables, where one variable decreases exponentially as the other variable increases. In other words, the function models a situation where a quantity decreases at a rate proportional to its current value.

Q: What are the characteristics of an exponential decay function?

A: Exponential decay functions have several key characteristics that distinguish them from other types of functions. These characteristics include:

• Exponential form: Exponential decay functions are typically written in the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable.
• Decaying rate: The rate at which the function decays is determined by the base $b$. If $b$ is between 0 and 1, the function decays exponentially. If $b$ is greater than 1, the function grows exponentially.
• Initial value: The initial value of the function is determined by the constant $a$. This is the value of the function when $x$ is equal to 0.

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

A: To determine if a function is an exponential decay function, you need to examine its form and characteristics. Specifically, you need to check if the function is in the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. You also need to check if the base $b$ is between 0 and 1, indicating that the function decays exponentially.

Q: What are some examples of exponential decay functions?

A: Here are some examples of exponential decay functions:

• $f(x) = 2(0.5)^x$
• $f(x) = 3(0.8)^{-x}$
• $f(x) = 4(0.9)^x$

Q: Can you provide more examples of exponential decay functions?

A: Here are some more examples of exponential decay functions:

• $f(x) = 5(0.7)^x$
• $f(x) = 6(0.6)^{-x}$
• $f(x) = 7(0.4)^x$

Q: How do I graph an exponential decay function?

A: To graph an exponential decay function, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph by hand. The graph of an exponential decay function will be a curve that decreases exponentially as the value of $x$ increases.

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

A: Exponential decay functions have many real-world applications, including:

• Radioactive decay: Exponential decay functions are used to model the decay of radioactive materials.
• Population growth: Exponential decay functions are used to model the decline of a population over time.
• Chemical reactions: Exponential decay functions are used to model the rate of chemical reactions.
• Economics: Exponential decay functions are used to model the decline of a company's stock price over time.

Q: Can you provide more information about the applications of exponential decay functions?

A: Here are some more examples of the applications of exponential decay functions:

• Biology: Exponential decay functions are used to model the decline of a species over time.
• Medicine: Exponential decay functions are used to model the decline of a disease over time.
• Finance: Exponential decay functions are used to model the decline of a company's stock price over time.
• Environmental science: Exponential decay functions are used to model the decline of a pollutant over time.

Conclusion

In conclusion, exponential decay functions are an important concept in mathematics and have many real-world applications. By understanding the characteristics and examples of exponential decay functions, you can better appreciate their importance and relevance in various fields.