What Is The Range Of The Function $y = 2e^x - 1$?A. All Real Numbers Less Than -1B. All Real Numbers Greater Than -1C. All Real Numbers Less Than 1D. All Real Numbers Greater Than 1

Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. When dealing with exponential functions, understanding the range is crucial to determine the behavior of the function and its possible output values. In this article, we will explore the range of the function $y = 2e^x - 1$ and discuss the possible answers.

What is an Exponential Function?

An exponential function is a function of the form $y = ab^x$, where $a$ and $b$ are constants, and $b$ is the base of the exponential function. The base $b$ can be any positive real number, but it is usually greater than 1. The function $y = 2e^x - 1$ is an example of an exponential function with base $e$, where $e$ is a mathematical constant approximately equal to 2.71828.

The Function $y = 2e^x - 1$

The function $y = 2e^x - 1$ is a simple exponential function with a base of $e$ and a coefficient of 2. The function can be rewritten as $y = 2e^x - 1 = 2(e^x) - 1$. This function has a horizontal asymptote at $y = -1$, which means that as $x$ approaches infinity, $y$ approaches -1.

Understanding the Range of Exponential Functions

The range of an exponential function is the set of all possible output values it can produce for the given input values. For the function $y = 2e^x - 1$, we can see that the function is always increasing as $x$ increases. This means that the function will always produce output values greater than -1.

Analyzing the Possible Answers

Now that we have a good understanding of the function $y = 2e^x - 1$, let's analyze the possible answers:

• A. All real numbers less than -1: This is not possible because the function is always increasing and will always produce output values greater than -1.
• B. All real numbers greater than -1: This is possible because the function is always increasing and will always produce output values greater than -1.
• C. All real numbers less than 1: This is not possible because the function will always produce output values greater than -1, which is greater than 1.
• D. All real numbers greater than 1: This is not possible because the function will always produce output values greater than -1, but not necessarily greater than 1.

Conclusion

In conclusion, the range of the function $y = 2e^x - 1$ is all real numbers greater than -1. This is because the function is always increasing and will always produce output values greater than -1.

Final Answer

Introduction

In our previous article, we explored the range of the function $y = 2e^x - 1$ and concluded that the range is all real numbers greater than -1. In this article, we will answer some frequently asked questions about the range of exponential functions and provide additional insights.

Q: What is the range of an exponential function?

A: The range of an exponential function is the set of all possible output values it can produce for the given input values. For the function $y = 2e^x - 1$, the range is all real numbers greater than -1.

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

A: To determine the range of an exponential function, you need to analyze the function's behavior as the input value approaches infinity. If the function is always increasing, the range will be all real numbers greater than the horizontal asymptote. If the function is always decreasing, the range will be all real numbers less than the horizontal asymptote.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the function approaches as the input value approaches infinity. For the function $y = 2e^x - 1$, the horizontal asymptote is $y = -1$.

Q: Can an exponential function have a range of all real numbers?

A: Yes, an exponential function can have a range of all real numbers if the base is greater than 1 and the coefficient is positive. For example, the function $y = 2e^x$ has a range of all real numbers.

Q: Can an exponential function have a range of all real numbers less than 0?

A: No, an exponential function cannot have a range of all real numbers less than 0. This is because the function will always produce output values greater than or equal to 0.

Q: How do I determine the range of a function with a base less than 1?

A: If the base of the exponential function is less than 1, the function will always decrease as the input value increases. In this case, the range will be all real numbers less than the horizontal asymptote.

Q: Can an exponential function have a range of all real numbers greater than 1?

A: Yes, an exponential function can have a range of all real numbers greater than 1 if the base is greater than 1 and the coefficient is positive. For example, the function $y = 2e^x$ has a range of all real numbers greater than 1.

Conclusion

In conclusion, understanding the range of exponential functions is crucial to determine the behavior of the function and its possible output values. By analyzing the function's behavior as the input value approaches infinity, you can determine the range of the function.

Final Tips

• Always analyze the function's behavior as the input value approaches infinity to determine the range.
• Use the horizontal asymptote to determine the lower bound of the range.
• Be careful when dealing with functions with a base less than 1, as the range may be all real numbers less than the horizontal asymptote.

Common Mistakes

• Assuming that the range of an exponential function is all real numbers without analyzing the function's behavior.
• Failing to consider the horizontal asymptote when determining the range.
• Not being careful when dealing with functions with a base less than 1.

Additional Resources

• Khan Academy: Exponential Functions
• Mathway: Exponential Functions
• Wolfram Alpha: Exponential Functions

Conclusion

In conclusion, understanding the range of exponential functions is crucial to determine the behavior of the function and its possible output values. By analyzing the function's behavior as the input value approaches infinity, you can determine the range of the function. Remember to always use the horizontal asymptote to determine the lower bound of the range and be careful when dealing with functions with a base less than 1.