Consider The Graph Of The Function F ( X ) = Ln ⁡ X F(x)=\ln X F ( X ) = Ln X .Which Is A Feature Of The Function G G G If G ( X ) = − F ( X − 4 G(x)=-f(x-4 G ( X ) = − F ( X − 4 ]?A. Vertical Asymptote At X = − 4 X=-4 X = − 4 B. Horizontal Asymptote At Y = 4 Y=4 Y = 4 C. Y Y Y -intercept At

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The function f(x)=ln⁡xf(x)=\ln x is a fundamental logarithmic function that has a wide range of applications in mathematics and other fields. The graph of this function is a fundamental concept in understanding the behavior of logarithmic functions. In this article, we will explore the graph of the function f(x)=ln⁡xf(x)=\ln x and use it to determine the features of the function g(x)=−f(x−4)g(x)=-f(x-4).

Graph of the Function f(x)=ln⁡xf(x)=\ln x

The graph of the function f(x)=ln⁡xf(x)=\ln x is a continuous and increasing function that has a vertical asymptote at x=0x=0. This means that as xx approaches 0 from the right, the value of f(x)f(x) approaches negative infinity. The graph of the function f(x)=ln⁡xf(x)=\ln x is also a one-to-one function, which means that it passes the horizontal line test.

Properties of the Graph of f(x)=ln⁡xf(x)=\ln x

The graph of the function f(x)=ln⁡xf(x)=\ln x has several important properties that are essential in understanding the behavior of logarithmic functions. Some of the key properties of the graph of f(x)=ln⁡xf(x)=\ln x include:

  • Domain: The domain of the function f(x)=ln⁡xf(x)=\ln x is all positive real numbers, i.e., x>0x>0.
  • Range: The range of the function f(x)=ln⁡xf(x)=\ln x is all real numbers, i.e., y∈Ry\in\mathbb{R}.
  • Vertical Asymptote: The graph of the function f(x)=ln⁡xf(x)=\ln x has a vertical asymptote at x=0x=0.
  • Horizontal Asymptote: The graph of the function f(x)=ln⁡xf(x)=\ln x has a horizontal asymptote at y=−∞y=-\infty as xx approaches 0 from the right.

Understanding the Function g(x)=−f(x−4)g(x)=-f(x-4)

The function g(x)=−f(x−4)g(x)=-f(x-4) is a transformation of the function f(x)=ln⁡xf(x)=\ln x. This transformation involves shifting the graph of the function f(x)=ln⁡xf(x)=\ln x to the right by 4 units and then reflecting it about the x-axis.

Properties of the Graph of g(x)=−f(x−4)g(x)=-f(x-4)

The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has several important properties that are essential in understanding the behavior of the function. Some of the key properties of the graph of g(x)=−f(x−4)g(x)=-f(x-4) include:

  • Domain: The domain of the function g(x)=−f(x−4)g(x)=-f(x-4) is all positive real numbers, i.e., x>4x>4.
  • Range: The range of the function g(x)=−f(x−4)g(x)=-f(x-4) is all real numbers, i.e., y∈Ry\in\mathbb{R}.
  • Vertical Asymptote: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a vertical asymptote at x=4x=4.
  • Horizontal Asymptote: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a horizontal asymptote at y=4y=4 as xx approaches infinity.

Determining the Features of the Function g(x)=−f(x−4)g(x)=-f(x-4)

Based on the properties of the graph of the function g(x)=−f(x−4)g(x)=-f(x-4), we can determine the features of the function. Some of the key features of the function g(x)=−f(x−4)g(x)=-f(x-4) include:

  • Vertical Asymptote at x=4x=4: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a vertical asymptote at x=4x=4.
  • Horizontal Asymptote at y=4y=4: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a horizontal asymptote at y=4y=4 as xx approaches infinity.
  • yy-Intercept at (4,0)(4,0): The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a yy-intercept at (4,0)(4,0).

Conclusion

In conclusion, the graph of the function f(x)=ln⁡xf(x)=\ln x is a fundamental concept in understanding the behavior of logarithmic functions. The function g(x)=−f(x−4)g(x)=-f(x-4) is a transformation of the function f(x)=ln⁡xf(x)=\ln x that involves shifting the graph to the right by 4 units and then reflecting it about the x-axis. The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has several important properties that are essential in understanding the behavior of the function. Some of the key features of the function g(x)=−f(x−4)g(x)=-f(x-4) include a vertical asymptote at x=4x=4, a horizontal asymptote at y=4y=4, and a yy-intercept at (4,0)(4,0).

References

  • [1] Calculus by Michael Spivak
  • [2] Calculus by James Stewart
  • [3] Mathematics by Michael Artin
    Q&A: Understanding the Graph of the Function f(x)=ln⁡xf(x)=\ln x and the Function g(x)=−f(x−4)g(x)=-f(x-4) =====================================================================================

In the previous article, we explored the graph of the function f(x)=ln⁡xf(x)=\ln x and the function g(x)=−f(x−4)g(x)=-f(x-4). In this article, we will answer some frequently asked questions about the graph of the function f(x)=ln⁡xf(x)=\ln x and the function g(x)=−f(x−4)g(x)=-f(x-4).

Q: What is the domain of the function f(x)=ln⁡xf(x)=\ln x?

A: The domain of the function f(x)=ln⁡xf(x)=\ln x is all positive real numbers, i.e., x>0x>0.

Q: What is the range of the function f(x)=ln⁡xf(x)=\ln x?

A: The range of the function f(x)=ln⁡xf(x)=\ln x is all real numbers, i.e., y∈Ry\in\mathbb{R}.

Q: What is the vertical asymptote of the function f(x)=ln⁡xf(x)=\ln x?

A: The graph of the function f(x)=ln⁡xf(x)=\ln x has a vertical asymptote at x=0x=0.

Q: What is the horizontal asymptote of the function f(x)=ln⁡xf(x)=\ln x?

A: The graph of the function f(x)=ln⁡xf(x)=\ln x has a horizontal asymptote at y=−∞y=-\infty as xx approaches 0 from the right.

Q: What is the domain of the function g(x)=−f(x−4)g(x)=-f(x-4)?

A: The domain of the function g(x)=−f(x−4)g(x)=-f(x-4) is all positive real numbers, i.e., x>4x>4.

Q: What is the range of the function g(x)=−f(x−4)g(x)=-f(x-4)?

A: The range of the function g(x)=−f(x−4)g(x)=-f(x-4) is all real numbers, i.e., y∈Ry\in\mathbb{R}.

Q: What is the vertical asymptote of the function g(x)=−f(x−4)g(x)=-f(x-4)?

A: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a vertical asymptote at x=4x=4.

Q: What is the horizontal asymptote of the function g(x)=−f(x−4)g(x)=-f(x-4)?

A: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a horizontal asymptote at y=4y=4 as xx approaches infinity.

Q: What is the yy-intercept of the function g(x)=−f(x−4)g(x)=-f(x-4)?

A: The graph of the function g(x)=−f(x−4)g(x)=-f(x-4) has a yy-intercept at (4,0)(4,0).

Q: How does the function g(x)=−f(x−4)g(x)=-f(x-4) differ from the function f(x)=ln⁡xf(x)=\ln x?

A: The function g(x)=−f(x−4)g(x)=-f(x-4) is a transformation of the function f(x)=ln⁡xf(x)=\ln x that involves shifting the graph to the right by 4 units and then reflecting it about the x-axis.

Q: What are some real-world applications of the function f(x)=ln⁡xf(x)=\ln x and the function g(x)=−f(x−4)g(x)=-f(x-4)?

A: The function f(x)=ln⁡xf(x)=\ln x and the function g(x)=−f(x−4)g(x)=-f(x-4) have several real-world applications in fields such as engineering, economics, and computer science. Some examples include:

  • Modeling population growth and decay
  • Analyzing financial data and predicting stock prices
  • Developing algorithms for computer graphics and game development

Conclusion

In conclusion, the graph of the function f(x)=ln⁡xf(x)=\ln x and the function g(x)=−f(x−4)g(x)=-f(x-4) are fundamental concepts in understanding the behavior of logarithmic functions. The Q&A section above provides answers to some frequently asked questions about the graph of the function f(x)=ln⁡xf(x)=\ln x and the function g(x)=−f(x−4)g(x)=-f(x-4). We hope that this article has provided a better understanding of these concepts and their real-world applications.

References

  • [1] Calculus by Michael Spivak
  • [2] Calculus by James Stewart
  • [3] Mathematics by Michael Artin