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

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 $y=4e^x$ and discuss the possible answers.

What is an Exponential Function?

An exponential function is a mathematical function of the form $y=a^x$, where $a$ is a positive constant and $x$ is the variable. The function $y=4e^x$ is an example of an exponential function, where $a=e$ (the base of the natural logarithm) and the coefficient is 4. The function $y=4e^x$ can be rewritten as $y=4e^{x}$, where $e$ is the base of the natural logarithm.

The Nature of Exponential Functions

Exponential functions have a unique property: they grow rapidly as the input value increases. In the case of the function $y=4e^x$, the output value increases exponentially as the input value $x$ increases. This means that as $x$ gets larger, the output value $y$ will get larger and larger.

The Range of Exponential Functions

The range of an exponential function is the set of all possible output values it can produce. For the function $y=4e^x$, the range is all real numbers greater than 0. This is because the function $y=4e^x$ is always positive, and as $x$ gets larger, the output value $y$ will get larger and larger.

Why is the Range of $y=4e^x$ All Real Numbers Greater than 0?

The reason why the range of $y=4e^x$ is all real numbers greater than 0 is because the function $y=4e^x$ is always positive. This is due to the fact that the base of the natural logarithm, $e$, is always positive, and the coefficient 4 is also positive. As a result, the output value $y$ will always be positive, and as $x$ gets larger, the output value $y$ will get larger and larger.

Why is the Range of $y=4e^x$ Not All Real Numbers Less than 0?

The range of $y=4e^x$ is not all real numbers less than 0 because the function $y=4e^x$ is always positive. This is due to the fact that the base of the natural logarithm, $e$, is always positive, and the coefficient 4 is also positive. As a result, the output value $y$ will always be positive, and as $x$ gets larger, the output value $y$ will get larger and larger.

Why is the Range of $y=4e^x$ Not All Real Numbers Less than 4?

The range of $y=4e^x$ is not all real numbers less than 4 because the function $y=4e^x$ is always greater than or equal to 4. This is due to the fact that the coefficient 4 is multiplied by the exponential function $e^x$, which is always greater than or equal to 1. As a result, the output value $y$ will always be greater than or equal to 4, and as $x$ gets larger, the output value $y$ will get larger and larger.

Why is the Range of $y=4e^x$ Not All Real Numbers Greater than 4?

The range of $y=4e^x$ is not all real numbers greater than 4 because the function $y=4e^x$ is always greater than or equal to 4. This is due to the fact that the coefficient 4 is multiplied by the exponential function $e^x$, which is always greater than or equal to 1. As a result, the output value $y$ will always be greater than or equal to 4, and as $x$ gets larger, the output value $y$ will get larger and larger.

Conclusion

In conclusion, the range of the function $y=4e^x$ is all real numbers greater than 0. This is because the function $y=4e^x$ is always positive, and as $x$ gets larger, the output value $y$ will get larger and larger. The range of $y=4e^x$ is not all real numbers less than 0, all real numbers less than 4, or all real numbers greater than 4 because the function $y=4e^x$ is always greater than or equal to 4.

Answer

The correct answer is A. All real numbers greater than 0.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Range of Exponential Functions" by Wolfram MathWorld

Additional Resources

• [1] "Exponential Functions" by Khan Academy
• [2] "Range of Exponential Functions" by MIT OpenCourseWare
  Q&A: Understanding the Range of Exponential Functions =====================================================

Introduction

In our previous article, we explored the range of the function $y=4e^x$ and discussed the possible answers. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

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 $y=4e^x$ all real numbers greater than 0?

A: The range of $y=4e^x$ is all real numbers greater than 0 because the function $y=4e^x$ is always positive. This is due to the fact that the base of the natural logarithm, $e$, is always positive, and the coefficient 4 is also positive. As a result, the output value $y$ will always be positive, and as $x$ gets larger, the output value $y$ will get larger and larger.

Q: Why is the range of $y=4e^x$ not all real numbers less than 0?

A: The range of $y=4e^x$ is not all real numbers less than 0 because the function $y=4e^x$ is always positive. This is due to the fact that the base of the natural logarithm, $e$, is always positive, and the coefficient 4 is also positive. As a result, the output value $y$ will always be positive, and as $x$ gets larger, the output value $y$ will get larger and larger.

Q: Why is the range of $y=4e^x$ not all real numbers less than 4?

A: The range of $y=4e^x$ is not all real numbers less than 4 because the function $y=4e^x$ is always greater than or equal to 4. This is due to the fact that the coefficient 4 is multiplied by the exponential function $e^x$, which is always greater than or equal to 1. As a result, the output value $y$ will always be greater than or equal to 4, and as $x$ gets larger, the output value $y$ will get larger and larger.

Q: Why is the range of $y=4e^x$ not all real numbers greater than 4?

A: The range of $y=4e^x$ is not all real numbers greater than 4 because the function $y=4e^x$ is always greater than or equal to 4. This is due to the fact that the coefficient 4 is multiplied by the exponential function $e^x$, which is always greater than or equal to 1. As a result, the output value $y$ will always be greater than or equal to 4, and as $x$ gets larger, the output value $y$ will get larger and larger.

Q: Can the range of $y=4e^x$ be expressed as an interval?

A: Yes, the range of $y=4e^x$ can be expressed as an interval. The interval is $(0, \infty)$, which represents all real numbers greater than 0.

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

A: The range of $y=e^x$ is also all real numbers greater than 0. However, the range of $y=4e^x$ is a vertical stretch of the range of $y=e^x$ by a factor of 4.

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

A: The domain of the function $y=4e^x$ is all real numbers. This is because the function $y=4e^x$ is defined for all real values of $x$.

Conclusion

In conclusion, the range of the function $y=4e^x$ is all real numbers greater than 0. This is due to the fact that the function $y=4e^x$ is always positive, and as $x$ gets larger, the output value $y$ will get larger and larger. We hope this Q&A section has helped clarify any doubts and provided additional information on the topic.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Range of Exponential Functions" by Wolfram MathWorld

Additional Resources

• [1] "Exponential Functions" by Khan Academy
• [2] "Range of Exponential Functions" by MIT OpenCourseWare