Not considering the domain and range of the functions

Oct 13, 2025 by windailys.com 44 views

Introduction

In mathematics, functions are used to describe relationships 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 x

where ln x is the natural logarithm of x. This function is a continuous and increasing function for all x > 0.

Function g(x)

The function g(x) is defined as:

g(x) = -5 ln x

This function is also a continuous and increasing function for all x > 0.

Comparing the Graphs

To compare the graphs of f(x) and g(x), we need to consider their properties. The graph of f(x) is a continuous and increasing curve that passes through the point (1, 0). The graph of g(x) is also a continuous and increasing curve, but it is steeper than the graph of f(x).

Domain and Range

The domain of f(x) is all x > 0, and the range is all real numbers. The domain of g(x) is also all x > 0, but the range is all real numbers multiplied by -5.

Behavior

The behavior of f(x) and g(x) is similar, but g(x) is steeper than f(x). This means that for a given value of x, g(x) will be less than f(x).

Key Differences

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

Scale: The graph of g(x) is steeper than the graph of f(x).
Range: The range of g(x) is all real numbers multiplied by -5, while the range of f(x) is all real numbers.
Behavior: For a given value of x, g(x) will be less than f(x).

Conclusion

In conclusion, the graph of function g(x) is steeper than the graph of function f(x). The range of g(x) is all real numbers multiplied by -5, while the range of f(x) is all real numbers. The behavior of g(x) is similar to f(x), but g(x) is less than f(x) for a given value of x.

Key Takeaways

The graph of g(x) is steeper than the graph of f(x).
The range of g(x) is all real numbers multiplied by -5.
The behavior of g(x) is similar to f(x), but g(x) is less than f(x) for a given value of x.

Final Thoughts

In this article, we compared the graphs of two functions, f(x) and g(x). We discussed their properties, such as domain, range, and behavior. We also highlighted the key differences between the graphs of f(x) and g(x. By understanding these differences, we can better analyze and compare the graphs of functions in mathematics.

Step-by-Step Solution

To solve this problem, follow these steps:

Understand the functions: Read and understand the definitions of f(x) and g(x).
Compare the graphs: Compare the properties of f(x) and g(x), such as domain, range, and behavior.
Highlight key differences: Identify the key differences between the graphs of f(x) and g(x).
Draw conclusions: Draw conclusions based on the comparison of the graphs of f(x) and g(x.

Example Questions

What is the domain of f(x)?
What is the range of g(x)?
How does the graph of g(x) compare with the graph of f(x)?

Answer Key

The domain of f(x) is all x > 0.
The range of g(x) is all real numbers multiplied by -5.
The graph of g(x) is steeper than the graph of f(x).
Q&A: Comparing the Graphs of Functions f and g =====================================================

Introduction

In our previous article, we compared the graphs of two functions, f(x) and g(x). We discussed their properties, such as domain, range, and behavior, and highlighted the key differences between the graphs of f(x) and g(x). In this article, we will answer some frequently asked questions about comparing the graphs of functions.

Q: What is the domain of f(x)?

A: The domain of f(x) is all x > 0.

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

A: The range of g(x) is all real numbers multiplied by -5.

Q: How does the graph of g(x) compare with the graph of f(x)?

A: The graph of g(x) is steeper than the graph of f(x).

Q: What is the behavior of g(x) compared to f(x)?

A: For a given value of x, g(x) will be less than f(x).

Q: What are the key differences between the graphs of f(x) and g(x)?

A: The key differences between the graphs of f(x) and g(x) are:

Scale: The graph of g(x) is steeper than the graph of f(x).
Range: The range of g(x) is all real numbers multiplied by -5, while the range of f(x) is all real numbers.
Behavior: For a given value of x, g(x) will be less than f(x).

Q: How can I determine the domain and range of a function?

A: To determine the domain and range of a function, you need to consider the properties of the function, such as the values of x that make the function undefined or undefined at certain points.

Q: What is the significance of comparing the graphs of functions?

A: Comparing the graphs of functions is important in mathematics because it helps us understand the behavior of functions and how they relate to each other.

Q: How can I apply the concepts of comparing graphs to real-world problems?

A: You can apply the concepts of comparing graphs to real-world problems by analyzing the behavior of functions in different contexts, such as economics, physics, or engineering.

Q: What are some common mistakes to avoid when comparing graphs?

A: Some common mistakes to avoid when comparing graphs include:

Not considering the domain and range of the functions
Not analyzing the behavior of the functions
Not highlighting the key differences between the graphs