The Range Of The Function Given Below Is The Set Of All Positive Real Numbers Greater Than 6.${ F(x) = 6 + 2^x }$A. True B. False

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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 discuss the range of the function F(x) = 6 + 2^x and determine whether the given statement is true or false.

Understanding the Function

The function F(x) = 6 + 2^x is an exponential function, where the base is 2 and the exponent is x. This function is defined for all real numbers x. To understand the range of this function, we need to analyze its behavior as x varies.

Analyzing the Function

As x increases, the value of 2^x also increases exponentially. When we add 6 to 2^x, the result is also an increasing function. However, the rate of increase is not constant, as it depends on the value of x.

Finding the Minimum Value

To find the minimum value of the function F(x) = 6 + 2^x, we need to find the smallest possible value of x. Since the function is defined for all real numbers x, the smallest possible value of x is -∞. However, as x approaches -∞, the value of 2^x approaches 0. Therefore, the minimum value of the function F(x) = 6 + 2^x is 6 + 0 = 6.

Finding the Maximum Value

To find the maximum value of the function F(x) = 6 + 2^x, we need to find the largest possible value of x. Since the function is defined for all real numbers x, the largest possible value of x is ∞. As x approaches ∞, the value of 2^x approaches ∞. Therefore, the maximum value of the function F(x) = 6 + 2^x is 6 + ∞ = ∞.

Conclusion

From the analysis above, we can conclude that the range of the function F(x) = 6 + 2^x is the set of all real numbers greater than 6. This is because the minimum value of the function is 6, and the maximum value is ∞. Therefore, the given statement is True.

Final Answer

The final answer is A. True.

Discussion

The range of a function is an important concept in mathematics, as it helps us understand the behavior of the function and its possible output values. In this article, we discussed the range of the function F(x) = 6 + 2^x and determined that the given statement is true. We also analyzed the behavior of the function and found that the minimum value is 6 and the maximum value is ∞.

Related Topics

  • Range of a function
  • Exponential functions
  • Mathematical analysis

References

  • [1] Calculus, James Stewart, 8th edition
  • [2] Algebra and Trigonometry, Michael Sullivan, 7th edition
  • [3] Mathematics for the Nonmathematician, Morris Kline, 1st edition

Introduction

In our previous article, we discussed the range of the function F(x) = 6 + 2^x and determined that the given statement is true. In this article, we will answer some frequently asked questions related to the range of this function.

Q1: What is the range of the function F(x) = 6 + 2^x?

A1: The range of the function F(x) = 6 + 2^x is the set of all real numbers greater than 6.

Q2: Why is the range of the function F(x) = 6 + 2^x greater than 6?

A2: The range of the function F(x) = 6 + 2^x is greater than 6 because the minimum value of the function is 6, and the maximum value is ∞.

Q3: Can the function F(x) = 6 + 2^x take on any value greater than 6?

A3: Yes, the function F(x) = 6 + 2^x can take on any value greater than 6. As x increases, the value of 2^x also increases exponentially, and when we add 6 to 2^x, the result is also an increasing function.

Q4: Is the function F(x) = 6 + 2^x a one-to-one function?

A4: No, the function F(x) = 6 + 2^x is not a one-to-one function. This is because the function is increasing, but it is not strictly increasing. There are multiple values of x that produce the same output value.

Q5: Can we find the inverse of the function F(x) = 6 + 2^x?

A5: Yes, we can find the inverse of the function F(x) = 6 + 2^x. However, the inverse function is not a simple function, and it is not easy to find.

Q6: What is the domain of the function F(x) = 6 + 2^x?

A6: The domain of the function F(x) = 6 + 2^x is the set of all real numbers.

Q7: Can we graph the function F(x) = 6 + 2^x?

A7: Yes, we can graph the function F(x) = 6 + 2^x. The graph of the function is an increasing exponential curve.

Q8: Is the function F(x) = 6 + 2^x continuous?

A8: Yes, the function F(x) = 6 + 2^x is continuous. This is because the function is defined for all real numbers x, and the function is differentiable at all points.

Q9: Can we find the derivative of the function F(x) = 6 + 2^x?

A9: Yes, we can find the derivative of the function F(x) = 6 + 2^x. The derivative of the function is F'(x) = 2^x * ln(2).

Q10: What is the second derivative of the function F(x) = 6 + 2^x?

A10: The second derivative of the function F(x) = 6 + 2^x is F''(x) = 2^x * (ln(2))^2.

Conclusion

In this article, we answered some frequently asked questions related to the range of the function F(x) = 6 + 2^x. We discussed the range of the function, the behavior of the function, and some of its properties.

Final Answer

The final answer is A1: The range of the function F(x) = 6 + 2^x is the set of all real numbers greater than 6.

Discussion

The range of a function is an important concept in mathematics, as it helps us understand the behavior of the function and its possible output values. In this article, we discussed the range of the function F(x) = 6 + 2^x and answered some frequently asked questions related to the range of this function.

Related Topics

  • Range of a function
  • Exponential functions
  • Mathematical analysis
  • Graphing functions
  • Derivatives
  • Second derivatives

References

  • [1] Calculus, James Stewart, 8th edition
  • [2] Algebra and Trigonometry, Michael Sullivan, 7th edition
  • [3] Mathematics for the Nonmathematician, Morris Kline, 1st edition