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. $f(x) =

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Exponential decay functions are a crucial concept in mathematics, particularly in calculus and algebra. These functions describe a phenomenon where a quantity decreases at a rate proportional to its current value. In this article, we will explore the characteristics of exponential decay functions and determine which of the given options represents such a function.

What is an Exponential Decay Function?

An exponential decay function is a mathematical function that describes a situation where a quantity decreases at a rate proportional to its current value. This type of function is often represented in the form:

f(x) = ab^x

where a is the initial value, b is the decay factor, and x is the variable. The decay factor (b) is a constant between 0 and 1, indicating the rate at which the quantity decreases.

Characteristics of Exponential Decay Functions

Exponential decay functions have several key characteristics:

  • Initial Value: The initial value (a) is the starting point of the function, representing the quantity at time x = 0.
  • Decay Factor: The decay factor (b) is a constant between 0 and 1, indicating the rate at which the quantity decreases.
  • Exponential Decrease: The function decreases exponentially as x increases, meaning that the quantity decreases at a rate proportional to its current value.
  • Asymptotic Behavior: As x approaches infinity, the function approaches 0, indicating that the quantity will eventually decrease to 0.

Analyzing the Options

Now that we have a clear understanding of exponential decay functions, let's analyze the given options:

Option A: f(x)=34(74)xf(x) = \frac{3}{4}\left(\frac{7}{4}\right)^x

This function can be rewritten as:

f(x) = \frac{3}{4} \cdot \left(\frac{7}{4}\right)^x

The decay factor (b) is 74\frac{7}{4}, which is greater than 1. This indicates that the function is actually an exponential growth function, not an exponential decay function.

Option B: f(x)=23(45)−xf(x) = \frac{2}{3}\left(\frac{4}{5}\right)^{-x}

This function can be rewritten as:

f(x) = \frac{2}{3} \cdot \left(\frac{5}{4}\right)^x

The decay factor (b) is 54\frac{5}{4}, which is greater than 1. This indicates that the function is actually an exponential growth function, not an exponential decay function.

Option C: f(x)=32(87)−xf(x) = \frac{3}{2}\left(\frac{8}{7}\right)^{-x}

This function can be rewritten as:

f(x) = \frac{3}{2} \cdot \left(\frac{7}{8}\right)^x

The decay factor (b) is 78\frac{7}{8}, which is between 0 and 1. This indicates that the function is an exponential decay function.

Option D: f(x)=32(87)xf(x) = \frac{3}{2}\left(\frac{8}{7}\right)^x

This function can be rewritten as:

f(x) = \frac{3}{2} \cdot \left(\frac{7}{8}\right)^{-x}

The decay factor (b) is 78\frac{7}{8}, which is between 0 and 1. This indicates that the function is an exponential decay function.

Conclusion

Based on our analysis, we can conclude that both Option C and Option D represent exponential decay functions. However, the correct answer is Option C: f(x)=32(87)−xf(x) = \frac{3}{2}\left(\frac{8}{7}\right)^{-x}.

Final Answer

Q&A: Exponential Decay Functions

In our previous article, we explored the concept of exponential decay functions and determined which of the given options represented such a function. In this article, we will provide a comprehensive guide to exponential decay functions, including a Q&A section to help you better understand this concept.

What is an Exponential Decay Function?

An exponential decay function is a mathematical function that describes a situation where a quantity decreases at a rate proportional to its current value. This type of function is often represented in the form:

f(x) = ab^x

where a is the initial value, b is the decay factor, and x is the variable. The decay factor (b) is a constant between 0 and 1, indicating the rate at which the quantity decreases.

Q: What is the initial value in an exponential decay function?

A: The initial value (a) is the starting point of the function, representing the quantity at time x = 0.

Q: What is the decay factor in an exponential decay function?

A: The decay factor (b) is a constant between 0 and 1, indicating the rate at which the quantity decreases.

Q: What is the asymptotic behavior of an exponential decay function?

A: As x approaches infinity, the function approaches 0, indicating that the quantity will eventually decrease 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, look for the following characteristics:

  • The function is in the form f(x) = ab^x
  • The decay factor (b) is a constant between 0 and 1
  • The function decreases exponentially as x increases

Q: What are some common applications of exponential decay functions?

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

  • Radioactive decay
  • Population growth and decline
  • Chemical reactions
  • Financial modeling

Q: How do I graph an exponential decay function?

A: To graph an exponential decay function, follow these steps:

  1. Determine the initial value (a) and decay factor (b)
  2. Plot the point (0, a) on the graph
  3. Plot additional points using the equation f(x) = ab^x
  4. Connect the points to form a curve

Q: Can I use exponential decay functions to model real-world phenomena?

A: Yes, exponential decay functions can be used to model many real-world phenomena, including population growth and decline, chemical reactions, and financial modeling.

Conclusion

Exponential decay functions are a powerful tool for modeling real-world phenomena. By understanding the characteristics of these functions, you can apply them to a wide range of applications. We hope this article has provided you with a comprehensive guide to exponential decay functions and has helped you better understand this concept.

Final Answer

The final answer is that exponential decay functions are a crucial concept in mathematics, particularly in calculus and algebra. They describe a phenomenon where a quantity decreases at a rate proportional to its current value. By understanding the characteristics of these functions, you can apply them to a wide range of applications.

Commonly Asked Questions

  • What is an exponential decay function?
    • An exponential decay function is a mathematical function that describes a situation where a quantity decreases at a rate proportional to its current value.
  • What is the initial value in an exponential decay function?
    • The initial value (a) is the starting point of the function, representing the quantity at time x = 0.
  • What is the decay factor in an exponential decay function?
    • The decay factor (b) is a constant between 0 and 1, indicating the rate at which the quantity decreases.
  • What is the asymptotic behavior of an exponential decay function?
    • As x approaches infinity, the function approaches 0, indicating that the quantity will eventually decrease to 0.