What Is The Range Of The Function $y = 4e^x$?A. All Real Numbers Greater Than 0 B. All Real Numbers Less Than 0 C. All Real Numbers Less Than 4 D. All Real Numbers Greater Than 4

Introduction

When dealing with functions, understanding their range is crucial in mathematics. The range of a function is the set of all possible output values it can produce for the given input values. In this article, we will explore the range of the function $y = 4e^x$. This function is an exponential function, where $e$ is the base of the natural logarithm, approximately equal to 2.71828. The function $y = 4e^x$ is a transformation of the basic exponential function $y = e^x$, where the output is multiplied by 4.

Understanding Exponential Functions

Exponential functions have the form $y = a^x$, where $a$ is the base and $x$ is the exponent. The base $a$ can be any positive real number, and the exponent $x$ can be any real number. When $a > 1$, the function is increasing, and when $0 < a < 1$, the function is decreasing. The function $y = e^x$ is a special case of an exponential function, where the base $a$ is equal to $e$. This function is increasing and has a horizontal asymptote at $y = 0$.

The Function $y = 4e^x$

The function $y = 4e^x$ is a transformation of the basic exponential function $y = e^x$. The output of the function $y = e^x$ is multiplied by 4, resulting in a new function. This transformation affects the range of the function. Since the output of the function $y = e^x$ is always positive, multiplying it by 4 will also result in a positive output.

Finding the Range of the Function

To find the range of the function $y = 4e^x$, we need to consider the possible output values it can produce. Since the function is an exponential function, it will always produce positive output values. The function $y = 4e^x$ will produce output values greater than 0, and there is no upper bound to these values. As $x$ approaches negative infinity, the output of the function $y = 4e^x$ approaches 0, but it will never reach 0. Therefore, the range of the function $y = 4e^x$ is all real numbers greater than 0.

Conclusion

In conclusion, the range of the function $y = 4e^x$ is all real numbers greater than 0. This is because the function is an exponential function, and the output is always positive. The transformation of the basic exponential function $y = e^x$ by multiplying the output by 4 results in a new function with a range of all real numbers greater than 0.

Final Answer

The final answer to the question "What is the range of the function $y = 4e^x$?" is:

A. All real numbers greater than 0

This is the correct answer, as the function $y = 4e^x$ will always produce positive output values, and there is no upper bound to these values.

Introduction

In our previous article, we explored the range of the function $y = 4e^x$. We found that the range of this function is all real numbers greater than 0. In this article, we will answer some frequently asked questions about the range of the function $y = 4e^x$.

Q&A

Q: What is the range of the function $y = 4e^x$?

A: The range of the function $y = 4e^x$ is all real numbers greater than 0.

Q: Why is the range of the function $y = 4e^x$ all real numbers greater than 0?

A: The range of the function $y = 4e^x$ is all real numbers greater than 0 because the function is an exponential function, and the output is always positive. The transformation of the basic exponential function $y = e^x$ by multiplying the output by 4 results in a new function with a range of all real numbers greater than 0.

Q: What happens to the output of the function $y = 4e^x$ as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the output of the function $y = 4e^x$ approaches 0, but it will never reach 0.

Q: Is there an upper bound to the output of the function $y = 4e^x$?

A: No, there is no upper bound to the output of the function $y = 4e^x$. The function will always produce output values greater than 0, and these values can become arbitrarily large.

Q: Can the function $y = 4e^x$ produce output values less than 0?

A: No, the function $y = 4e^x$ will never produce output values less than 0. The function is an exponential function, and the output is always positive.

Q: How does the range of the function $y = 4e^x$ compare to the range of the function $y = e^x$?

A: The range of the function $y = 4e^x$ is the same as the range of the function $y = e^x$, which is all real numbers greater than 0. However, the function $y = 4e^x$ will produce output values that are 4 times larger than the output values of the function $y = e^x$.

Conclusion

In conclusion, the range of the function $y = 4e^x$ is all real numbers greater than 0. This is because the function is an exponential function, and the output is always positive. We hope that this Q&A article has helped to clarify any questions you may have had about the range of the function $y = 4e^x$.

Final Answer

The final answer to the question "What is the range of the function $y = 4e^x$?" is:

A. All real numbers greater than 0

This is the correct answer, as the function $y = 4e^x$ will always produce positive output values, and there is no upper bound to these values.