Determine The Range Of An Exponential Decay Function In The Form $y = A B^x$ That Has An Initial Value Of 4 And A Decay Of 5%.A. All Real NumbersB. $x \ \textgreater \ 0$C. $y \ \textless \ 0$D. $y \ \textgreater \

Introduction

Exponential decay functions are a fundamental concept in mathematics, used to model various real-world phenomena, such as population growth, chemical reactions, and financial transactions. In this article, we will focus on determining the range of an exponential decay function in the form $y = a b^x$, where $a$ is the initial value and $b$ is the decay factor.

Understanding Exponential Decay Functions

An exponential decay function is a mathematical function that describes a process where a quantity decreases over time. The general form of an exponential decay function is $y = a b^x$, where:

• $y$ is the dependent variable (the quantity being measured)
• $a$ is the initial value (the value of $y$ at $x = 0$)
• $b$ is the decay factor (a constant between 0 and 1 that determines the rate of decay)
• $x$ is the independent variable (time or the variable that affects the decay)

The Initial Value and Decay Factor

In this problem, we are given that the initial value $a$ is 4 and the decay factor $b$ is 5% (or 0.05). This means that the function starts at $y = 4$ and decreases by 5% for each unit increase in $x$.

Determining the Range

To determine the range of the function, we need to find the values of $y$ that the function can take. Since the function is an exponential decay function, it will always decrease as $x$ increases. Therefore, the range of the function will be all values less than or equal to the initial value $a$.

The Correct Answer

Based on the above analysis, the correct answer is:

• D. $y \ \textless \ 0$

However, since the initial value $a$ is 4, the function will never take on values less than 0. Therefore, the correct answer is:

• D. $y \ \textless \ 4$

Conclusion

In conclusion, the range of an exponential decay function in the form $y = a b^x$ with an initial value of 4 and a decay of 5% is all values less than 4.

Range of Exponential Decay Functions

Exponential decay functions have a range that depends on the initial value and the decay factor. In general, the range of an exponential decay function is all values less than or equal to the initial value.

Examples of Exponential Decay Functions

Here are some examples of exponential decay functions:

• $y = 2(0.8)^x$ (initial value 2, decay factor 0.8)
• $y = 3(0.9)^x$ (initial value 3, decay factor 0.9)
• $y = 1(0.5)^x$ (initial value 1, decay factor 0.5)

Solving Exponential Decay Functions

To solve an exponential decay function, we need to isolate the variable $x$. This can be done using logarithms.

Logarithmic Form

The logarithmic form of an exponential decay function is:

• $\log(y) = \log(a) + x \log(b)$

Solving for $x$

To solve for $x$, we can use the following formula:

• $x = \frac{\log(y) - \log(a)}{\log(b)}$

Conclusion

In conclusion, exponential decay functions are a fundamental concept in mathematics, used to model various real-world phenomena. The range of an exponential decay function depends on the initial value and the decay factor. By understanding the properties of exponential decay functions, we can solve problems involving these functions.

References

• [1] "Exponential Decay Functions" by Math Is Fun
• [2] "Exponential Decay" by Wolfram MathWorld
• [3] "Logarithmic Form" by Khan Academy
  Exponential Decay Functions: Q&A =====================================

Introduction

Exponential decay functions are a fundamental concept in mathematics, used to model various real-world phenomena, such as population growth, chemical reactions, and financial transactions. In this article, we will answer some frequently asked questions about exponential decay functions.

Q: What is an exponential decay function?

A: An exponential decay function is a mathematical function that describes a process where a quantity decreases over time. The general form of an exponential decay function is $y = a b^x$, where:

• $y$ is the dependent variable (the quantity being measured)
• $a$ is the initial value (the value of $y$ at $x = 0$)
• $b$ is the decay factor (a constant between 0 and 1 that determines the rate of decay)
• $x$ is the independent variable (time or the variable that affects the decay)

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

A: The initial value $a$ is the value of $y$ at $x = 0$. It is the starting point of the function.

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

A: The decay factor $b$ is a constant between 0 and 1 that determines the rate of decay. A value of $b$ close to 1 means a slow decay, while a value of $b$ close to 0 means a fast decay.

Q: How do I determine the range of an exponential decay function?

A: To determine the range of an exponential decay function, you need to find the values of $y$ that the function can take. Since the function is an exponential decay function, it will always decrease as $x$ increases. Therefore, the range of the function will be all values less than or equal to the initial value $a$.

Q: Can an exponential decay function take on negative values?

A: No, an exponential decay function cannot take on negative values. Since the function is an exponential decay function, it will always decrease as $x$ increases, and the minimum value it can take is 0.

Q: How do I solve an exponential decay function?

A: To solve an exponential decay function, you need to isolate the variable $x$. This can be done using logarithms.

Q: What is the logarithmic form of an exponential decay function?

A: The logarithmic form of an exponential decay function is:

• $\log(y) = \log(a) + x \log(b)$

Q: How do I solve for $x$ in an exponential decay function?

A: To solve for $x$, you can use the following formula:

• $x = \frac{\log(y) - \log(a)}{\log(b)}$

Q: What are some examples of exponential decay functions?

A: Here are some examples of exponential decay functions:

• $y = 2(0.8)^x$ (initial value 2, decay factor 0.8)
• $y = 3(0.9)^x$ (initial value 3, decay factor 0.9)
• $y = 1(0.5)^x$ (initial value 1, decay factor 0.5)

Conclusion

In conclusion, exponential decay functions are a fundamental concept in mathematics, used to model various real-world phenomena. By understanding the properties of exponential decay functions, we can solve problems involving these functions.

References

• [1] "Exponential Decay Functions" by Math Is Fun
• [2] "Exponential Decay" by Wolfram MathWorld
• [3] "Logarithmic Form" by Khan Academy