Understanding Growth and Decay Functions

Oct 13, 2025 by magicalceylontours.com 45 views

Understanding Growth and Decay Functions

In mathematics, growth and decay functions are used to describe how a quantity changes over time. A growth function represents an increase in the quantity, while a decay function represents a decrease. Identifying whether a function represents growth or decay is crucial in various fields, including economics, biology, and physics.

The Given Function

The function given is $ f(t) = 300 \cdot 1.12^5 $. To determine whether this function represents growth or decay, we need to analyze its behavior.

Analyzing the Function

The function $ f(t) = 300 \cdot 1.12^5 $ is an exponential function, where the base is 1.12. To understand its behavior, let's break it down:

The coefficient 300 represents the initial value of the function.
The base 1.12 is greater than 1, indicating that the function will increase over time.
The exponent 5 represents the number of times the base is multiplied by itself.

Calculating the Function Value

To calculate the function value, we need to evaluate $ 300 \cdot 1.12^5 $. Using a calculator or a programming language, we get:

$ f(t) = 300 \cdot 1.12^5 \approx 300 \cdot 1.762 = 530.6 $

Comparing with the Options

The calculated function value is approximately 530.6, which is greater than the initial value of 300. This indicates that the function represents growth.

Conclusion

Based on the analysis, the function $ f(t) = 300 \cdot 1.12^5 $ represents growth. The base 1.12 is greater than 1, indicating an increase in the function value over time.

Growth or Decay: The Final Answer

The final answer is:

Growth: The function $ f(t) = 300 \cdot 1.12^5 $ represents growth.

Additional Discussion

In this discussion, we analyzed a simple exponential function to determine whether it represents growth or decay. The key takeaway is that an exponential function with a base greater than 1 represents growth, while a base less than 1 represents decay.

Real-World Applications

Growth and decay functions have numerous real-world applications, including:

Population growth and decline
Economic growth and recession
Chemical reactions and decay
Biological growth and decay

Conclusion

Frequently Asked Questions

In the previous article, we discussed how to identify whether a function represents growth or decay. Here are some frequently asked questions and answers related to growth and decay functions:

Q: What is the difference between a growth function and a decay function?

A: A growth function represents an increase in the quantity over time, while a decay function represents a decrease in the quantity over time.

Q: How do I determine whether a function represents growth or decay?

A: To determine whether a function represents growth or decay, you need to analyze its behavior. If the base of the function is greater than 1, it represents growth. If the base is less than 1, it represents decay.

Q: What is an example of a growth function?

A: An example of a growth function is $ f(t) = 300 \cdot 1.12^t $. In this function, the base 1.12 is greater than 1, indicating an increase in the function value over time.

Q: What is an example of a decay function?

A: An example of a decay function is $ f(t) = 300 \cdot 0.88^t $. In this function, the base 0.88 is less than 1, indicating a decrease in the function value over time.

Q: Can a function represent both growth and decay?

A: Yes, a function can represent both growth and decay, depending on the value of the base and the exponent. For example, the function $ f(t) = 300 \cdot 1.12^t $ represents growth, while the function $ f(t) = 300 \cdot 0.88^t $ represents decay.

Q: How do I calculate the value of an exponential function?

A: To calculate the value of an exponential function, you need to evaluate the expression using the base and exponent. For example, to calculate the value of $ f(t) = 300 \cdot 1.12^t $, you need to evaluate $ 300 \cdot 1.12^t $.

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

A: The exponent in an exponential function represents the number of times the base is multiplied by itself. For example, in the function $ f(t) = 300 \cdot 1.12^t $, the exponent t represents the number of times the base 1.12 is multiplied by itself.

Q: Can I use a calculator to calculate the value of an exponential function?

A: Yes, you can use a calculator to calculate the value of an exponential function. Most calculators have a built-in function for calculating exponential values.

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

A: Growth and decay functions have numerous real-world applications, including:

Population growth and decline
Economic growth and recession
Chemical reactions and decay
Biological growth and decay

Conclusion

In conclusion, growth and decay functions are used to describe how a quantity changes over time. By analyzing the function's behavior, we can determine whether it represents an increase or decrease in the quantity over time.