Statement a is always true

Oct 13, 2025 by windailys.com 21 views

Introduction

Exponential decay functions are a fundamental concept in mathematics, describing how quantities decrease over time. In this article, we will consider two exponential decay functions, ${$ f $}$ and ${$ g $}$, and determine whether certain statements are always, sometimes, or never true. We will justify our answers using mathematical reasoning and examples.

Statement a: ${$ (f \cdot g)(x) $}$ is an exponential decay function

To begin, let's consider the product of two exponential decay functions, ${$ (f \cdot g)(x) $}$. An exponential decay function has the general form ${$ f(x) = a \cdot e^{bx} $}$, where ${$ a $}$ and ${$ b $}$ are positive constants, and ${$ e $}$ is the base of the natural logarithm.

When we multiply two exponential decay functions, we get:

${$ (f \cdot g)(x) = (a \cdot e^{bx}) \cdot (c \cdot e^{dx}) $}$

where ${$ a $}$ and ${$ c $}$ are positive constants, and ${$ b $}$ and ${$ d $}$ are also positive constants.

Using the properties of exponents, we can simplify this expression:

${$ (f \cdot g)(x) = (a \cdot c) \cdot e^{(b+d)x} $}$

Now, let's examine the properties of this function. Since ${$ a \cdot c $}$ is a positive constant, and ${$ b + d $}$ is also a positive constant, the function ${$ (f \cdot g)(x) $}$ has the general form of an exponential decay function.

Therefore, we can conclude that:

Statement a is always true: The product of two exponential decay functions is also an exponential decay function.

Statement b: ${$ (f + g)(x) $}$ is an exponential decay function

Next, let's consider the sum of two exponential decay functions, ${$ (f + g)(x) $}$. Using the same general form for exponential decay functions, we can write:

${$ (f + g)(x) = (a \cdot e^{bx}) + (c \cdot e^{dx}) $}$

Now, let's examine the properties of this function. Since ${$ a $}$ and ${$ c $}$ are positive constants, and ${$ b $}$ and ${$ d $}$ are also positive constants, the function ${$ (f + g)(x) $}$ is a sum of two exponential decay functions.

However, the sum of two exponential decay functions is not necessarily an exponential decay function. In fact, the sum can be a linear function, a quadratic function, or even a more complex function.

To see why, let's consider an example. Suppose we have two exponential decay functions:

${$ f(x) = 2 \cdot e^{-x} $}$

${$ g(x) = 3 \cdot e^{-2x} $}$

When we add these two functions, we get:

${$ (f + g)(x) = 2 \cdot e^{-x} + 3 \cdot e^{-2x} $}$

This function is not an exponential decay function, but rather a linear function.

Therefore, we can conclude that:

Statement b is sometimes true: The sum of two exponential decay functions is not necessarily an exponential decay function.

Statement c: ${$ (f \cdot g)(x) $}$ is a faster decay function than ${$ f(x) $}$

Next, let's consider the product of two exponential decay functions, ${$ (f \cdot g)(x) $}$. We have already shown that this function is also an exponential decay function.

Now, let's compare the decay rates of ${$ (f \cdot g)(x) $}$ and ${$ f(x) $}$. Since ${$ (f \cdot g)(x) $}$ has a larger exponent, ${$ b + d $}$, it will decay faster than ${$ f(x) $}$.

To see why, let's consider an example. Suppose we have two exponential decay functions:

${$ f(x) = 2 \cdot e^{-x} $}$

${$ g(x) = 3 \cdot e^{-2x} $}$

When we multiply these two functions, we get:

${$ (f \cdot g)(x) = 6 \cdot e^{-3x} $}$

This function decays faster than ${$ f(x) $}$, since it has a larger exponent.

Therefore, we can conclude that:

Statement c is always true: The product of two exponential decay functions is a faster decay function than either of the original functions.

Conclusion

In this article, we have considered two exponential decay functions, ${$ f $}$ and ${$ g $}$, and determined whether certain statements are always, sometimes, or never true. We have justified our answers using mathematical reasoning and examples.

We have shown that:

The product of two exponential decay functions is also an exponential decay function (Statement a is always true).
The sum of two exponential decay functions is not necessarily an exponential decay function (Statement b is sometimes true).
The product of two exponential decay functions is a faster decay function than either of the original functions (Statement c is always true).

Introduction

In our previous article, we explored the properties of exponential decay functions and determined whether certain statements are always, sometimes, or never true. In this article, we will provide a Q&A guide to help you better understand exponential decay functions and their applications.

Q: What is an exponential decay function?

A: An exponential decay function is a mathematical function that describes how a quantity decreases over time. It has the general form ${$ f(x) = a \cdot e^{bx} $}$, where ${$ a $}$ and ${$ b $}$ are positive constants, and ${$ e $}$ is the base of the natural logarithm.

Q: What are some examples of exponential decay functions?

A: Some examples of exponential decay functions include:

${$ f(x) = 2 \cdot e^{-x} $}$
${$ f(x) = 3 \cdot e^{-2x} $}$
${$ f(x) = 4 \cdot e^{-3x} $}$

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

A: An exponential growth function has the general form ${$ f(x) = a \cdot e^{bx} $}$, where ${$ a $}$ and ${$ b $}$ are positive constants. In contrast, an exponential decay function has the general form ${$ f(x) = a \cdot e^{-bx} $}$, where ${$ a $}$ and ${$ b $}$ are positive constants.

Q: How do you determine whether a function is an exponential decay function or not?

A: To determine whether a function is an exponential decay function, you can check if it has the general form ${$ f(x) = a \cdot e^{-bx} $}$, where ${$ a $}$ and ${$ b $}$ are positive constants. If it does, then it is an exponential decay function.

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

A: The exponent in an exponential decay function determines the rate at which the quantity decreases over time. A larger exponent means a faster decay rate, while a smaller exponent means a slower decay rate.

Q: Can you give an example of how to use exponential decay functions in real-world applications?

A: Yes, exponential decay functions are used in many real-world applications, such as:

Modeling the decay of radioactive materials
Describing the spread of diseases
Analyzing the growth of populations
Predicting the behavior of financial markets

For example, suppose we want to model the decay of a radioactive substance with a half-life of 10 years. We can use an exponential decay function to describe the decay over time:

${$ f(x) = 100 \cdot e^{-0.1x} $}$

where ${$ x $}$ is the number of years since the substance was created.

Q: How do you calculate the half-life of an exponential decay function?

A: To calculate the half-life of an exponential decay function, you can use the formula:

${$ t_{1/2} = \frac{\ln(2)}{b} $}$

where ${$ t_{1/2} $}$ is the half-life, ${$ \ln(2) $}$ is the natural logarithm of 2, and ${$ b $}$ is the exponent in the exponential decay function.

For example, suppose we have an exponential decay function:

${$ f(x) = 100 \cdot e^{-0.1x} $}$

To calculate the half-life, we can use the formula:

${$ t_{1/2} = \frac{\ln(2)}{0.1} = 6.93 $}$

Therefore, the half-life of this substance is approximately 6.93 years.

Conclusion

In this article, we have provided a Q&A guide to help you better understand exponential decay functions and their applications. We have covered topics such as the definition of exponential decay functions, examples of exponential decay functions, and real-world applications of exponential decay functions. We hope that this guide has been helpful in your understanding of exponential decay functions.