Select All The Correct Answers.Consider Function $f$ And Function $g$.${ \begin{align*} f(x) &= \ln X \ g(x) &= -5 \ln X \end{align*} }$How Does The Graph Of Function $g$ Compare With The Graph Of Function

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Introduction

In mathematics, functions are used to describe the relationship between variables. When comparing the graphs of two functions, we need to consider their properties, such as domain, range, and behavior. In this article, we will compare the graphs of two functions, f(x) and g(x), and discuss how they differ.

Function f(x)

The function f(x) is defined as:

f(x)=ln⁡xf(x) = \ln x

This is a natural logarithmic function, which is defined for all positive real numbers. The graph of f(x) is a curve that increases as x increases.

Function g(x)

The function g(x) is defined as:

g(x)=−5ln⁡xg(x) = -5 \ln x

This is a modified version of the natural logarithmic function, with a negative coefficient. The graph of g(x) will be a reflection of the graph of f(x) across the x-axis.

Comparing the Graphs

To compare the graphs of f(x) and g(x), we need to consider their properties. The domain of both functions is the set of all positive real numbers. The range of f(x) is all real numbers, while the range of g(x) is also all real numbers, but with a negative sign.

The graph of g(x) will be a reflection of the graph of f(x) across the x-axis. This means that if we take any point (x, y) on the graph of f(x), the corresponding point on the graph of g(x) will be (x, -y).

Key Differences

There are several key differences between the graphs of f(x) and g(x:

  • Reflection: The graph of g(x) is a reflection of the graph of f(x) across the x-axis.
  • Scale: The graph of g(x) is scaled by a factor of -5 compared to the graph of f(x).
  • Range: The range of g(x) is all real numbers, but with a negative sign.

Conclusion

In conclusion, the graph of function g(x) is a reflection of the graph of function f(x) across the x-axis, with a scale factor of -5 and a negative range. This comparison highlights the importance of considering the properties of functions when comparing their graphs.

Key Takeaways

  • The graph of g(x) is a reflection of the graph of f(x) across the x-axis.
  • The graph of g(x) is scaled by a factor of -5 compared to the graph of f(x).
  • The range of g(x) is all real numbers, but with a negative sign.

Further Reading

For further reading on functions and their graphs, we recommend the following resources:

References

Discussion

Introduction

In our previous article, we compared the graphs of two functions, f(x) and g(x), and discussed how they differ. In this article, we will answer some frequently asked questions about comparing the graphs of functions.

Q: What is the main difference between the graphs of f(x) and g(x)?

A: The main difference between the graphs of f(x) and g(x) is that the graph of g(x) is a reflection of the graph of f(x) across the x-axis. This means that if we take any point (x, y) on the graph of f(x), the corresponding point on the graph of g(x) will be (x, -y).

Q: How does the scale factor affect the graph of g(x)?

A: The scale factor of -5 affects the graph of g(x) by scaling it vertically. This means that the graph of g(x) will be stretched or compressed vertically by a factor of 5 compared to the graph of f(x).

Q: What is the range of the graph of g(x)?

A: The range of the graph of g(x) is all real numbers, but with a negative sign. This means that the graph of g(x) will always be below the x-axis.

Q: How does the domain of f(x) and g(x) affect their graphs?

A: The domain of both functions is the set of all positive real numbers. This means that the graphs of f(x) and g(x) will only be defined for positive values of x.

Q: Can we compare the graphs of f(x) and g(x) using other methods?

A: Yes, we can compare the graphs of f(x) and g(x) using other methods, such as:

  • Horizontal shift: We can shift the graph of f(x) horizontally to compare it with the graph of g(x).
  • Vertical shift: We can shift the graph of f(x) vertically to compare it with the graph of g(x).
  • Scaling: We can scale the graph of f(x) horizontally or vertically to compare it with the graph of g(x).

Q: How can we use the comparison of the graphs of f(x) and g(x) in real-world applications?

A: The comparison of the graphs of f(x) and g(x) can be used in real-world applications, such as:

  • Modeling population growth: We can use the graph of f(x) to model the growth of a population, and the graph of g(x) to model the decline of a population.
  • Analyzing economic data: We can use the graph of f(x) to analyze economic data, such as the growth of a company's revenue, and the graph of g(x) to analyze the decline of a company's revenue.

Q: What are some other functions that we can compare with f(x) and g(x)?

A: Some other functions that we can compare with f(x) and g(x) are:

  • Exponential functions: We can compare the graphs of exponential functions, such as f(x) = 2^x and g(x) = 2^(-x).
  • Logarithmic functions: We can compare the graphs of logarithmic functions, such as f(x) = log(x) and g(x) = -log(x).
  • Trigonometric functions: We can compare the graphs of trigonometric functions, such as f(x) = sin(x) and g(x) = -sin(x).

Conclusion

In conclusion, comparing the graphs of functions is an important concept in mathematics. By understanding how to compare the graphs of functions, we can gain a deeper understanding of the properties of functions and how they can be used to model real-world phenomena.

Key Takeaways

  • The graph of g(x) is a reflection of the graph of f(x) across the x-axis.
  • The graph of g(x) is scaled by a factor of -5 compared to the graph of f(x).
  • The range of g(x) is all real numbers, but with a negative sign.
  • We can compare the graphs of functions using other methods, such as horizontal shift, vertical shift, and scaling.
  • The comparison of the graphs of functions can be used in real-world applications, such as modeling population growth and analyzing economic data.