Write The Exponential Function That Describes The Situation:The Initial Amount Of The Function Is 19, And Then It Decreases By A Factor Of 3 7 \frac{3}{7} 7 3 ​ With Each Unit Increase In X X X .The Function That Describes The Situation Is

Introduction

In various real-world scenarios, we encounter situations where a quantity decreases exponentially with each unit increase in a particular variable. This phenomenon can be modeled using exponential functions, which are a fundamental concept in mathematics. In this article, we will explore the exponential function that describes a situation where the initial amount is 19, and then it decreases by a factor of $\frac{3}{7}$ with each unit increase in $x$.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, where one variable is a constant power of the other variable. The general form of an exponential function is $f(x) = ab^x$, where $a$ is the initial amount, $b$ is the base, and $x$ is the variable. In this case, we are interested in a decreasing exponential function, where the amount decreases with each unit increase in $x$.

The Exponential Function Describing the Situation

Given that the initial amount is 19, and the amount decreases by a factor of $\frac{3}{7}$ with each unit increase in $x$, we can write the exponential function as:

$$f(x) = 19 \left(\frac{3}{7}\right)^x$$

This function describes the situation where the amount decreases exponentially with each unit increase in $x$. The base of the exponential function is $\frac{3}{7}$, which represents the factor by which the amount decreases with each unit increase in $x$.

Properties of the Exponential Function

The exponential function $f(x) = 19 \left(\frac{3}{7}\right)^x$ has several important properties that are worth noting:

• Domain: The domain of the function is all real numbers, $x \in \mathbb{R}$.
• Range: The range of the function is all positive real numbers, $f(x) \in (0, \infty)$.
• Asymptotes: The function has a horizontal asymptote at $y = 0$, which means that as $x$ approaches infinity, the function approaches 0.
• Increasing/Decreasing: The function is decreasing for all values of $x$, since the base is less than 1.

Graphing the Exponential Function

To visualize the exponential function, we can graph it using a graphing calculator or a computer algebra system. The graph of the function is a decreasing exponential curve that approaches the horizontal asymptote at $y = 0$ as $x$ approaches infinity.

Applications of the Exponential Function

The exponential function $f(x) = 19 \left(\frac{3}{7}\right)^x$ has several real-world applications, including:

• Population modeling: The function can be used to model the population of a species that decreases exponentially with each unit increase in time.
• Financial modeling: The function can be used to model the value of an investment that decreases exponentially with each unit increase in time.
• Chemical reactions: The function can be used to model the concentration of a chemical that decreases exponentially with each unit increase in time.

Conclusion

In conclusion, the exponential function $f(x) = 19 \left(\frac{3}{7}\right)^x$ describes a situation where the initial amount is 19, and then it decreases by a factor of $\frac{3}{7}$ with each unit increase in $x$. The function has several important properties, including a domain of all real numbers, a range of all positive real numbers, and a horizontal asymptote at $y = 0$. The function has several real-world applications, including population modeling, financial modeling, and chemical reactions.

Introduction

In our previous article, we explored the exponential function that describes a situation where the initial amount is 19, and then it decreases by a factor of $\frac{3}{7}$ with each unit increase in $x$. In this article, we will answer some frequently asked questions related to the exponential function and its applications.

Q&A

Q1: What is the initial amount of the exponential function?

A1: The initial amount of the exponential function is 19.

Q2: What is the base of the exponential function?

A2: The base of the exponential function is $\frac{3}{7}$, which represents the factor by which the amount decreases with each unit increase in $x$.

Q3: What is the domain of the exponential function?

A3: The domain of the exponential function is all real numbers, $x \in \mathbb{R}$.

Q4: What is the range of the exponential function?

A4: The range of the exponential function is all positive real numbers, $f(x) \in (0, \infty)$.

Q5: What is the horizontal asymptote of the exponential function?

A5: The horizontal asymptote of the exponential function is $y = 0$, which means that as $x$ approaches infinity, the function approaches 0.

Q6: Is the exponential function increasing or decreasing?

A6: The exponential function is decreasing for all values of $x$, since the base is less than 1.

Q7: How can the exponential function be used in real-world applications?

A7: The exponential function can be used in various real-world applications, including population modeling, financial modeling, and chemical reactions.

Q8: Can the exponential function be used to model a situation where the amount increases with each unit increase in $x$?

A8: No, the exponential function we discussed in this article is a decreasing exponential function, which means that the amount decreases with each unit increase in $x$. If you want to model a situation where the amount increases with each unit increase in $x$, you would need to use an increasing exponential function.

Q9: How can the exponential function be graphed?

A9: The exponential function can be graphed using a graphing calculator or a computer algebra system.

Q10: What are some common mistakes to avoid when working with exponential functions?

A10: Some common mistakes to avoid when working with exponential functions include:

• Not checking the domain and range of the function: Make sure to check the domain and range of the function to ensure that it is valid for the given values of $x$.
• Not checking the horizontal asymptote of the function: Make sure to check the horizontal asymptote of the function to ensure that it is valid for the given values of $x$.
• Not using the correct base and exponent: Make sure to use the correct base and exponent when working with exponential functions.

Conclusion

In conclusion, the exponential function $f(x) = 19 \left(\frac{3}{7}\right)^x$ describes a situation where the initial amount is 19, and then it decreases by a factor of $\frac{3}{7}$ with each unit increase in $x$. We have answered some frequently asked questions related to the exponential function and its applications. We hope that this article has been helpful in understanding the exponential function and its properties.