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

Mar 12, 2025 by ADMIN 214 views

Comparing the Graphs of Functions f(x) and g(x)

When comparing the graphs of two functions, it's essential to understand the properties and characteristics of each function. In this discussion, we will compare the graphs of functions f(x) and g(x), where f(x) = ln x and g(x) = -5 ln x. We will analyze the key differences between the two functions and how they affect the graph of g(x) compared to f(x).

Before we dive into the comparison, let's understand the properties of each function.

Function f(x) = ln x: The natural logarithm function, denoted as ln x, is a continuous and increasing function for all x > 0. The graph of f(x) is a curve that starts at negative infinity and increases as x approaches positive infinity.
Function g(x) = -5 ln x: The function g(x) is a transformation of the natural logarithm function f(x). The negative sign in front of the function indicates a reflection across the x-axis, while the coefficient -5 scales the function vertically.

Now that we understand the properties of each function, let's compare the graphs of f(x) and g(x).

Reflection across the x-axis: The graph of g(x) is a reflection of the graph of f(x) across the x-axis. This means that for every point (x, y) on the graph of f(x), there is a corresponding point (x, -y) on the graph of g(x).
Vertical scaling: The graph of g(x) is also scaled vertically by a factor of 5. This means that the graph of g(x) is 5 times taller than the graph of f(x).
Shift in the y-axis: Since the graph of g(x) is a reflection and scaling of the graph of f(x), it is also shifted in the y-axis. The graph of g(x) is shifted down by 5 units compared to the graph of f(x).

To further illustrate the comparison between the graphs of f(x) and g(x), let's consider some example problems.

Problem 1: If the graph of f(x) passes through the point (2, 1), what is the corresponding point on the graph of g(x)?
Solution: Since the graph of g(x) is a reflection of the graph of f(x) across the x-axis, the corresponding point on the graph of g(x) is (2, -1).
Problem 2: If the graph of f(x) has a maximum value of 2 at x = 1, what is the maximum value of the graph of g(x)?
Solution: Since the graph of g(x) is scaled vertically by a factor of 5, the maximum value of the graph of g(x) is 10.

In conclusion, the graph of function g(x) = -5 ln x is a reflection of the graph of function f(x) = ln x across the x-axis, scaled vertically by a factor of 5, and shifted down by 5 units in the y-axis. Understanding these transformations is essential in comparing the graphs of two functions and analyzing their properties. By applying these concepts, we can solve example problems and gain a deeper understanding of the properties of functions.
Q&A: Comparing the Graphs of Functions f(x) and g(x)

In this Q&A section, we will address some common questions related to comparing the graphs of functions f(x) and g(x).

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, scaled vertically by a factor of 5, and shifted down by 5 units in the y-axis.

Q: How does the reflection across the x-axis affect the graph of g(x)?

A: The reflection across the x-axis means that for every point (x, y) on the graph of f(x), there is a corresponding point (x, -y) on the graph of g(x). This means that the graph of g(x) is a mirror image of the graph of f(x) across the x-axis.

Q: What is the effect of the vertical scaling on the graph of g(x)?

A: The vertical scaling by a factor of 5 means that the graph of g(x) is 5 times taller than the graph of f(x). This means that the graph of g(x) will have the same shape as the graph of f(x), but it will be stretched vertically by a factor of 5.

Q: How does the shift in the y-axis affect the graph of g(x)?

A: The shift down by 5 units in the y-axis means that the graph of g(x) is shifted down by 5 units compared to the graph of f(x). This means that the graph of g(x) will have the same shape as the graph of f(x), but it will be shifted down by 5 units.

Q: Can you provide an example of how to compare the graphs of f(x) and g(x)?

A: Let's consider an example where the graph of f(x) passes through the point (2, 1). To find the corresponding point on the graph of g(x), we need to reflect the point (2, 1) across the x-axis, scale it vertically by a factor of 5, and shift it down by 5 units. The corresponding point on the graph of g(x) is (2, -1).

Q: How can I use this knowledge to solve problems involving the comparison of graphs?

A: To solve problems involving the comparison of graphs, you can use the following steps:

Identify the key differences between the graphs of f(x) and g(x).
Reflect the graph of f(x) across the x-axis to get the graph of g(x).
Scale the graph of g(x) vertically by a factor of 5.
Shift the graph of g(x) down by 5 units in the y-axis.
Use the resulting graph to solve the problem.

Q: Are there any other transformations that can be applied to the graph of f(x) to get the graph of g(x)?

A: Yes, there are other transformations that can be applied to the graph of f(x) to get the graph of g(x). Some of these transformations include:

Horizontal scaling: This involves scaling the graph of f(x) horizontally by a factor of k.
Horizontal shifting: This involves shifting the graph of f(x) horizontally by a distance of k units.
Vertical shifting: This involves shifting the graph of f(x) vertically by a distance of k units.