What Is The Range Of The Function $y = E^{4x}$?A. $y \ \textless \ 0$B. \$y \ \textgreater \ 0$[/tex\]C. $y \ \textless \ 4$D. $y \ \textgreater \ 4$

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Introduction

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 $y = e^{4x}$, where $e$ is the base of the natural logarithm and $x$ is the input variable.

Understanding the Function

The function $y = e^{4x}$ is an exponential function, where the base is $e$ and the exponent is $4x$. This means that for every input value of $x$, the output value of $y$ will be the result of raising $e$ to the power of $4x$. The function is continuous and differentiable for all real values of $x$.

Analyzing the Function

To determine the range of the function, we need to analyze its behavior as $x$ varies. Since the exponent $4x$ is a linear function of $x$, we can rewrite the function as $y = e^{4x} = (e4)x$. This shows that the function is an exponential function with base $e^4$.

Properties of Exponential Functions

Exponential functions have several important properties that can help us determine their range. One of the key properties is that exponential functions are always positive, except when the base is negative and the exponent is an even integer. In this case, the base $e^4$ is positive, so the function $y = e^{4x}$ is always positive.

Determining the Range

Since the function $y = e^{4x}$ is always positive, we can conclude that the range of the function is the set of all positive real numbers. In other words, for any real value of $x$, the output value of $y$ will be a positive real number.

Conclusion

In conclusion, the range of the function $y = e^{4x}$ is the set of all positive real numbers. This means that for any real value of $x$, the output value of $y$ will be a positive real number.

Final Answer

Based on our analysis, the correct answer is:

  • B. $y \ \textgreater \ 0$

This answer is supported by the fact that the function $y = e^{4x}$ is always positive, except when the base is negative and the exponent is an even integer. In this case, the base $e^4$ is positive, so the function $y = e^{4x}$ is always positive.

Additional Information

  • The range of the function $y = e^{4x}$ is the set of all positive real numbers.
  • The function $y = e^{4x}$ is always positive, except when the base is negative and the exponent is an even integer.
  • The base $e^4$ is positive, so the function $y = e^{4x}$ is always positive.

References

Related Topics

  • Exponential Functions
  • Range of a Function
  • Properties of Exponential Functions

Introduction

In our previous article, we explored the range of the function $y = e^{4x}$ and concluded that the range is the set of all positive real numbers. In this article, we will answer some frequently asked questions related to the range of the function $y = e^{4x}$.

Q&A

Q1: What is the range of the function $y = e^{4x}$?

A1: The range of the function $y = e^{4x}$ is the set of all positive real numbers.

Q2: Why is the function $y = e^{4x}$ always positive?

A2: The function $y = e^{4x}$ is always positive because the base $e^4$ is positive. Exponential functions with positive bases are always positive.

Q3: Can the function $y = e^{4x}$ ever be negative?

A3: No, the function $y = e^{4x}$ can never be negative. As we mentioned earlier, exponential functions with positive bases are always positive.

Q4: What happens when the exponent $4x$ is an even integer?

A4: When the exponent $4x$ is an even integer, the function $y = e^{4x}$ is still positive. However, if the base were negative, the function would be negative.

Q5: Can the function $y = e^{4x}$ be zero?

A5: No, the function $y = e^{4x}$ can never be zero. Exponential functions with positive bases are always positive, and zero is not a positive number.

Q6: What is the relationship between the range of the function $y = e^{4x}$ and the domain of the function?

A6: The range of the function $y = e^{4x}$ is the set of all positive real numbers, and the domain of the function is all real numbers. This means that for any real value of $x$, the output value of $y$ will be a positive real number.

Q7: Can the function $y = e^{4x}$ be undefined?

A7: No, the function $y = e^{4x}$ is never undefined. Exponential functions are defined for all real numbers, and the function $y = e^{4x}$ is no exception.

Q8: What is the significance of the range of the function $y = e^{4x}$?

A8: The range of the function $y = e^{4x}$ is significant because it tells us what values the function can take on. In this case, the function can take on any positive real number.

Conclusion

In conclusion, the range of the function $y = e^{4x}$ is the set of all positive real numbers. This means that for any real value of $x$, the output value of $y$ will be a positive real number. We hope this Q&A article has helped to clarify any questions you may have had about the range of the function $y = e^{4x}$.

Final Answer

Based on our analysis, the correct answer is:

  • B. $y \ \textgreater \ 0$

This answer is supported by the fact that the function $y = e^{4x}$ is always positive, except when the base is negative and the exponent is an even integer. In this case, the base $e^4$ is positive, so the function $y = e^{4x}$ is always positive.

Additional Information

  • The range of the function $y = e^{4x}$ is the set of all positive real numbers.
  • The function $y = e^{4x}$ is always positive, except when the base is negative and the exponent is an even integer.
  • The base $e^4$ is positive, so the function $y = e^{4x}$ is always positive.

References

Related Topics

  • Exponential Functions
  • Range of a Function
  • Properties of Exponential Functions