Which Is An Exponential Decay Function?A. { F(x) = 0.2\left(\frac{5}{3}\right)^x $}$ B. { F(x) = -3.1^x $}$ C. { F(x) = 10^x $}$ D. { F(x) = -2(0.7)^x $}$

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Exponential decay functions are a crucial concept in mathematics, particularly in calculus and algebra. These functions describe how a quantity decreases over time, with the rate of decrease being proportional to the 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 how a quantity decreases over time. It is characterized by the following properties:

  • The function has a base value, which is the value that the function approaches as time increases.
  • The function decreases exponentially, meaning that the rate of decrease is proportional to the current value.
  • The function can be represented in the form f(x) = ab^x, where a is the initial value, b is the decay rate, and x is the time variable.

Characteristics of Exponential Decay Functions

Exponential decay functions have several key characteristics that distinguish them from other types of functions. Some of the most important characteristics include:

  • Exponential decrease: The function decreases exponentially, meaning that the rate of decrease is proportional to the current value.
  • Base value: The function approaches a base value as time increases.
  • Decay rate: The function has a decay rate, which determines how quickly the function decreases.
  • Time variable: The function is typically represented in terms of a time variable, such as x.

Analyzing the Given Options

Now that we have a good understanding of exponential decay functions, let's analyze the given options to determine which one represents such a function.

Option A: { f(x) = 0.2\left(\frac{5}{3}\right)^x $}$

This function has a base value of 0.2 and a decay rate of 5/3. However, the decay rate is not a fraction between 0 and 1, which is typically the case for exponential decay functions. Therefore, this option does not represent an exponential decay function.

Option B: { f(x) = -3.1^x $}$

This function has a base value of -3.1 and a decay rate of 1. However, the base value is negative, which is not typical for exponential decay functions. Additionally, the decay rate is not a fraction between 0 and 1, which is typically the case for exponential decay functions. Therefore, this option does not represent an exponential decay function.

Option C: { f(x) = 10^x $}$

This function has a base value of 10 and a decay rate of 1. However, the base value is not a fraction between 0 and 1, which is typically the case for exponential decay functions. Additionally, the function does not decrease exponentially, but rather increases exponentially. Therefore, this option does not represent an exponential decay function.

Option D: { f(x) = -2(0.7)^x $}$

This function has a base value of -2 and a decay rate of 0.7. The decay rate is a fraction between 0 and 1, which is typical for exponential decay functions. Additionally, the function decreases exponentially, as required for an exponential decay function. Therefore, this option represents an exponential decay function.

Conclusion

In conclusion, the correct answer is Option D: { f(x) = -2(0.7)^x $}$. This function represents an exponential decay function, with a base value of -2 and a decay rate of 0.7. The function decreases exponentially, as required for an exponential decay function.

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Additional Resources

In our previous article, we explored the characteristics of exponential decay functions and determined which of the given options represented such a function. In this article, we will answer some frequently asked questions about exponential decay functions.

Q: What is the difference between exponential decay and exponential growth?

A: Exponential decay and exponential growth are two types of exponential functions that describe how a quantity changes over time. Exponential decay describes how a quantity decreases over time, while exponential growth describes how a quantity increases over time.

Q: What is the formula for an exponential decay function?

A: The formula for an exponential decay function is f(x) = ab^x, where a is the initial value, b is the decay rate, and x is the time variable.

Q: What is the significance of the decay rate in an exponential decay function?

A: The decay rate is a crucial component of an exponential decay function, as it determines how quickly the function decreases. A decay rate of 1 represents a linear decrease, while a decay rate between 0 and 1 represents an exponential decrease.

Q: Can an exponential decay function have a negative base value?

A: Yes, an exponential decay function can have a negative base value. However, the function will still decrease exponentially, as required for an exponential decay function.

Q: Can an exponential decay function have a decay rate greater than 1?

A: No, an exponential decay function cannot have a decay rate greater than 1. A decay rate greater than 1 would represent an exponential growth function, rather than an exponential decay function.

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 decreases exponentially, meaning that the rate of decrease is proportional to the current value.
  • The function has a base value, which is the value that the function approaches as time increases.
  • The function has a decay rate, which determines how quickly the function decreases.

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

A: Exponential decay functions have numerous 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.

Q: How do I graph an exponential decay function?

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

  1. Determine the base value and decay rate of the function.
  2. Choose a value for x, the time variable.
  3. Calculate the corresponding value of the function using the formula f(x) = ab^x.
  4. Plot the point (x, f(x)) on a coordinate plane.
  5. Repeat steps 2-4 for multiple values of x to create a graph of the function.

Conclusion

In conclusion, exponential decay functions are a crucial concept in mathematics, particularly in calculus and algebra. By understanding the characteristics of exponential decay functions and how to determine if a function is an exponential decay function, you can apply this knowledge to a variety of real-world problems.

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Additional Resources