Select All The Correct Answers.Which Statements Are True About Function { G $} ? ? ? [ G(x) = \left{ \begin{array}{ll} \left(\frac{1}{2}\right)^{x-2}, & X \ \textless \ 2 \ x^3 - 9x^2 + 27x - 25, & X \geq 2 \end{array} \right.

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Understanding the Function g(x)

Overview of the Function g(x)

The function g(x) is a piecewise function, which means it has different definitions for different intervals of x. In this case, the function has two definitions:

  • For x < 2, the function is defined as g(x) = (1/2)^(x-2)
  • For x ≥ 2, the function is defined as g(x) = x^3 - 9x^2 + 27x - 25

Properties of the Function g(x)

To determine which statements are true about the function g(x), we need to analyze its properties. Here are some key properties to consider:

  • Domain: The domain of a function is the set of all possible input values (x) for which the function is defined. In this case, the domain of g(x) is all real numbers, since both definitions are defined for all real numbers.
  • Range: The range of a function is the set of all possible output values (g(x)) for which the function is defined. We need to analyze the range of both definitions separately.
  • Continuity: A function is continuous if it can be drawn without lifting the pencil from the paper. We need to check if the function is continuous at x = 2, where the two definitions meet.
  • Differentiability: A function is differentiable if it has a derivative at every point. We need to check if the function is differentiable at x = 2.

Analysis of the Function g(x)

Let's analyze the function g(x) in more detail.

Definition 1: g(x) = (1/2)^(x-2)

  • Domain: The domain of this definition is all real numbers x < 2.
  • Range: The range of this definition is all positive real numbers, since (1/2)^(x-2) is always positive.
  • Continuity: This definition is continuous for all x < 2.
  • Differentiability: This definition is differentiable for all x < 2.

Definition 2: g(x) = x^3 - 9x^2 + 27x - 25

  • Domain: The domain of this definition is all real numbers x ≥ 2.
  • Range: The range of this definition is all real numbers, since x^3 - 9x^2 + 27x - 25 can take on any real value.
  • Continuity: This definition is continuous for all x ≥ 2.
  • Differentiability: This definition is differentiable for all x ≥ 2.

Continuity and Differentiability at x = 2

To determine if the function is continuous and differentiable at x = 2, we need to check if the two definitions meet at this point.

  • Continuity: The two definitions meet at x = 2, since (1/2)^(2-2) = 1 and 2^3 - 9(2)^2 + 27(2) - 25 = 1. Therefore, the function is continuous at x = 2.
  • Differentiability: To check if the function is differentiable at x = 2, we need to check if the derivative of the two definitions meet at this point. The derivative of (1/2)^(x-2) is -ln(1/2)(1/2)^(x-2), and the derivative of x^3 - 9x^2 + 27x - 25 is 3x^2 - 18x + 27. Evaluating these derivatives at x = 2, we get -ln(1/2)(1/2)^0 = -ln(1/2) and 3(2)^2 - 18(2) + 27 = 3. Therefore, the function is not differentiable at x = 2.

Conclusion

Based on the analysis above, we can conclude that the following statements are true about the function g(x):

  • The function g(x) is defined for all real numbers.
  • The function g(x) is continuous at x = 2.
  • The function g(x) is not differentiable at x = 2.

The following statements are false about the function g(x):

  • The function g(x) is differentiable at x = 2.
  • The function g(x) has a range of all real numbers for x < 2.
  • The function g(x) has a range of all positive real numbers for x ≥ 2.

Final Answer

The correct answers are:

  • The function g(x) is defined for all real numbers.
  • The function g(x) is continuous at x = 2.
  • The function g(x) is not differentiable at x = 2.

The incorrect answers are:

  • The function g(x) is differentiable at x = 2.
  • The function g(x) has a range of all real numbers for x < 2.
  • The function g(x) has a range of all positive real numbers for x ≥ 2.
    Q&A: Understanding the Function g(x)

Frequently Asked Questions

We've received many questions about the function g(x) and its properties. Here are some of the most frequently asked questions and their answers:

Q: What is the domain of the function g(x)?

A: The domain of the function g(x) is all real numbers, since both definitions are defined for all real numbers.

Q: What is the range of the function g(x) for x < 2?

A: The range of the function g(x) for x < 2 is all positive real numbers, since (1/2)^(x-2) is always positive.

Q: What is the range of the function g(x) for x ≥ 2?

A: The range of the function g(x) for x ≥ 2 is all real numbers, since x^3 - 9x^2 + 27x - 25 can take on any real value.

Q: Is the function g(x) continuous at x = 2?

A: Yes, the function g(x) is continuous at x = 2, since the two definitions meet at this point.

Q: Is the function g(x) differentiable at x = 2?

A: No, the function g(x) is not differentiable at x = 2, since the derivatives of the two definitions do not meet at this point.

Q: What is the derivative of the function g(x) for x < 2?

A: The derivative of the function g(x) for x < 2 is -ln(1/2)(1/2)^(x-2).

Q: What is the derivative of the function g(x) for x ≥ 2?

A: The derivative of the function g(x) for x ≥ 2 is 3x^2 - 18x + 27.

Q: Can the function g(x) be simplified to a single expression?

A: No, the function g(x) cannot be simplified to a single expression, since it has two different definitions for different intervals of x.

Q: Is the function g(x) a polynomial function?

A: No, the function g(x) is not a polynomial function, since it has a non-polynomial definition for x < 2.

Q: Is the function g(x) a rational function?

A: No, the function g(x) is not a rational function, since it has a non-rational definition for x < 2.

Additional Resources

If you have any further questions about the function g(x) or its properties, please don't hesitate to ask. We have a list of additional resources that may be helpful:

  • Mathematical References: For a more in-depth understanding of the function g(x) and its properties, please refer to the following mathematical references:
  • Calculus: A First Course by James Stewart
  • Calculus: Early Transcendentals by James Stewart
  • A First Course in Calculus by Serge Lang
  • Online Resources: For a more interactive understanding of the function g(x) and its properties, please visit the following online resources:
  • Khan Academy: Calculus
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Calculus

Conclusion

We hope this Q&A article has been helpful in understanding the function g(x) and its properties. If you have any further questions or need additional resources, please don't hesitate to ask.