Select The Correct Answer From Each Drop-down Menu.Complete The Statement To Correctly Describe The End Behavior Of The Given Function.$f(x)=10(0.75)^x$The Left End Approaches $\square$, And The Right End Approaches

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding their end behavior is crucial for solving various mathematical problems. In this article, we will delve into the end behavior of the given function, $f(x)=10(0.75)^x$, and determine the correct answer from each drop-down menu.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that exhibits rapid growth or decay. The general form of an exponential function is $f(x)=ab^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the variable. In the given function, $f(x)=10(0.75)^x$, the base is $0.75$, and the initial value is $10$.

End Behavior of Exponential Functions

The end behavior of an exponential function refers to the behavior of the function as $x$ approaches positive or negative infinity. In the case of the given function, we need to determine the left and right end behavior.

Left End Behavior

As $x$ approaches negative infinity, the value of $f(x)$ approaches $0$. This is because the base, $0.75$, is less than $1$, and as $x$ becomes more negative, the value of $f(x)$ decreases rapidly.

Right End Behavior

As $x$ approaches positive infinity, the value of $f(x)$ approaches infinity. This is because the base, $0.75$, is less than $1$, and as $x$ becomes more positive, the value of $f(x)$ increases rapidly.

Conclusion

In conclusion, the left end of the function $f(x)=10(0.75)^x$ approaches $0$, and the right end approaches infinity. Therefore, the correct answer from each drop-down menu is:

• Left end approaches: 0
• Right end approaches: ∞

Key Takeaways

• Exponential functions exhibit rapid growth or decay.
• The end behavior of an exponential function refers to the behavior of the function as $x$ approaches positive or negative infinity.
• The left end of the function $f(x)=10(0.75)^x$ approaches $0$, and the right end approaches infinity.

Real-World Applications

Understanding the end behavior of exponential functions has numerous real-world applications, including:

• Modeling population growth or decline
• Analyzing the spread of diseases
• Predicting the behavior of financial markets
• Understanding the behavior of chemical reactions

Common Mistakes

When analyzing the end behavior of exponential functions, it's essential to avoid common mistakes, including:

• Assuming that the function will approach a specific value as $x$ approaches infinity.
• Failing to consider the base and initial value of the function.
• Not recognizing the rapid growth or decay of the function.

Tips for Success

To succeed in analyzing the end behavior of exponential functions, follow these tips:

• Understand the properties of exponential functions.
• Recognize the base and initial value of the function.
• Use graphical or numerical methods to visualize the end behavior of the function.
• Avoid common mistakes and assumptions.

Conclusion

In conclusion, understanding the end behavior of exponential functions is crucial for solving various mathematical problems. By recognizing the left and right end behavior of the function $f(x)=10(0.75)^x$, we can gain valuable insights into the behavior of the function. Remember to avoid common mistakes and follow the tips for success to excel in analyzing the end behavior of exponential functions.

Introduction

In our previous article, we explored the end behavior of the exponential function $f(x)=10(0.75)^x$. We determined that the left end approaches $0$ and the right end approaches infinity. In this article, we will address some of the most frequently asked questions related to the end behavior of exponential functions.

Q&A

Q: What is the end behavior of an exponential function with a base greater than 1?

A: The end behavior of an exponential function with a base greater than 1 is that the left end approaches $0$ and the right end approaches infinity.

Q: What is the end behavior of an exponential function with a base less than 1?

A: The end behavior of an exponential function with a base less than 1 is that the left end approaches infinity and the right end approaches $0$.

Q: How does the end behavior of an exponential function change when the base is between 0 and 1?

A: When the base is between 0 and 1, the end behavior of the exponential function is that the left end approaches infinity and the right end approaches $0$.

Q: Can the end behavior of an exponential function be determined by its graph?

A: Yes, the end behavior of an exponential function can be determined by its graph. As $x$ approaches positive or negative infinity, the graph of the function will approach a horizontal asymptote.

Q: How does the initial value of an exponential function affect its end behavior?

A: The initial value of an exponential function does not affect its end behavior. The end behavior is determined solely by the base of the function.

Q: Can the end behavior of an exponential function be determined by its equation?

A: Yes, the end behavior of an exponential function can be determined by its equation. By analyzing the base and initial value of the function, we can determine the left and right end behavior.

Q: What is the significance of the end behavior of an exponential function in real-world applications?

A: The end behavior of an exponential function is significant in real-world applications because it helps us understand the behavior of the function as $x$ approaches positive or negative infinity. This is crucial in modeling population growth or decline, analyzing the spread of diseases, predicting the behavior of financial markets, and understanding the behavior of chemical reactions.

Conclusion

In conclusion, understanding the end behavior of exponential functions is crucial for solving various mathematical problems. By recognizing the left and right end behavior of the function, we can gain valuable insights into the behavior of the function. Remember to avoid common mistakes and follow the tips for success to excel in analyzing the end behavior of exponential functions.

Key Takeaways

• The end behavior of an exponential function with a base greater than 1 is that the left end approaches $0$ and the right end approaches infinity.
• The end behavior of an exponential function with a base less than 1 is that the left end approaches infinity and the right end approaches $0$.
• The initial value of an exponential function does not affect its end behavior.
• The end behavior of an exponential function can be determined by its graph and equation.

Real-World Applications

Understanding the end behavior of exponential functions has numerous real-world applications, including:

• Modeling population growth or decline
• Analyzing the spread of diseases
• Predicting the behavior of financial markets
• Understanding the behavior of chemical reactions

Common Mistakes

When analyzing the end behavior of exponential functions, it's essential to avoid common mistakes, including:

• Assuming that the function will approach a specific value as $x$ approaches infinity.
• Failing to consider the base and initial value of the function.
• Not recognizing the rapid growth or decay of the function.

Tips for Success

To succeed in analyzing the end behavior of exponential functions, follow these tips:

• Understand the properties of exponential functions.
• Recognize the base and initial value of the function.
• Use graphical or numerical methods to visualize the end behavior of the function.
• Avoid common mistakes and assumptions.