What Is The Range Of The Function Y = 4 E X Y = 4 E^x Y = 4 E 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

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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 $y = 4 e^x$. This function is an exponential function, where $e$ is the base of the natural logarithm, approximately equal to 2.71828.

Understanding Exponential Functions

Exponential functions have the form $y = a e^x$, where $a$ is a constant and $e$ is the base of the natural logarithm. The graph of an exponential function is a curve that increases or decreases rapidly as $x$ increases or decreases. The rate of increase or decrease depends on the value of $a$. If $a$ is positive, the function increases as $x$ increases, and if $a$ is negative, the function decreases as $x$ increases.

The Function $y = 4 e^x$

The function $y = 4 e^x$ is a specific type of exponential function where $a = 4$. This means that the function will increase rapidly as $x$ increases. To find the range of this function, we need to consider the possible output values it can produce for the given input values.

Finding the Range

To find the range of the function $y = 4 e^x$, we need to consider the behavior of the function as $x$ approaches positive and negative infinity. As $x$ approaches positive infinity, $e^x$ approaches infinity, and therefore, $4 e^x$ also approaches infinity. This means that the function has no upper bound and can produce arbitrarily large output values.

On the other hand, as $x$ approaches negative infinity, $e^x$ approaches 0, and therefore, $4 e^x$ also approaches 0. This means that the function has a lower bound of 0 and can produce output values arbitrarily close to 0.

Conclusion

Based on the analysis above, we can conclude that the range of the function $y = 4 e^x$ is all real numbers greater than 0. This is because the function can produce arbitrarily large output values as $x$ approaches positive infinity, and it can produce output values arbitrarily close to 0 as $x$ approaches negative infinity.

Final Answer

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

A. All real numbers greater than 0

This is the correct answer because the function can produce arbitrarily large output values as $x$ approaches positive infinity, and it can produce output values arbitrarily close to 0 as $x$ approaches negative infinity.

Discussion

The range of a function is an important concept in mathematics, and understanding it is crucial in solving problems involving functions. In this article, we explored the range of the function $y = 4 e^x$ and found that it is all real numbers greater than 0. This is because the function can produce arbitrarily large output values as $x$ approaches positive infinity, and it can produce output values arbitrarily close to 0 as $x$ approaches negative infinity.

Related Topics

• Exponential functions
• Range of a function
• Limits of a function
• Graphing functions

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Range of a Function" by Khan Academy
• [3] "Limits of a Function" by Wolfram MathWorld

Additional Resources

• [1] "Exponential Functions" by MIT OpenCourseWare
• [2] "Range of a Function" by Purplemath
• [3] "Limits of a Function" by Mathway
  

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Introduction

In our previous article, we explored the range of the function $y = 4 e^x$ and found that it is all real numbers greater than 0. However, we understand that some readers may still have questions about this topic. In this article, we will address some of the most frequently asked questions about the range of the function $y = 4 e^x$.

Q&A

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

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

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

A: The range of the function $y = 4 e^x$ is all real numbers greater than 0 because the function can produce arbitrarily large output values as $x$ approaches positive infinity, and it can produce output values arbitrarily close to 0 as $x$ approaches negative infinity.

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

A: No, the function $y = 4 e^x$ cannot produce output values less than 0. This is because the function is an exponential function with a positive base, and it always produces positive output values.

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

A: No, the function $y = 4 e^x$ cannot produce output values equal to 0. This is because the function is an exponential function with a positive base, and it always produces positive output values.

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

A: The domain of the function $y = 4 e^x$ is all real numbers.

Q: Why is the domain of the function $y = 4 e^x$ all real numbers?

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

Q: Can the function $y = 4 e^x$ be defined for complex values of $x$?

A: Yes, the function $y = 4 e^x$ can be defined for complex values of $x$. However, the range of the function would be different for complex values of $x$.

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

A: The graph of the function $y = 4 e^x$ is a curve that increases rapidly as $x$ increases.

Q: Why does the graph of the function $y = 4 e^x$ increase rapidly as $x$ increases?

A: The graph of the function $y = 4 e^x$ increases rapidly as $x$ increases because the function is an exponential function with a positive base.

Conclusion

In this article, we addressed some of the most frequently asked questions about the range of the function $y = 4 e^x$. We hope that this article has provided a better understanding of the range of this function and has helped to clarify any confusion.

Final Answer

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

A. All real numbers greater than 0

This is the correct answer because the function can produce arbitrarily large output values as $x$ approaches positive infinity, and it can produce output values arbitrarily close to 0 as $x$ approaches negative infinity.

Discussion

The range of a function is an important concept in mathematics, and understanding it is crucial in solving problems involving functions. In this article, we explored the range of the function $y = 4 e^x$ and found that it is all real numbers greater than 0. This is because the function can produce arbitrarily large output values as $x$ approaches positive infinity, and it can produce output values arbitrarily close to 0 as $x$ approaches negative infinity.

Related Topics

• Exponential functions
• Range of a function
• Limits of a function
• Graphing functions

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Range of a Function" by Khan Academy
• [3] "Limits of a Function" by Wolfram MathWorld

Additional Resources

• [1] "Exponential Functions" by MIT OpenCourseWare
• [2] "Range of a Function" by Purplemath
• [3] "Limits of a Function" by Mathway