Understanding the Difference of Functions

Oct 13, 2025 by windailys.com 139 views

Introduction

In mathematics, the concept of intervals is crucial in understanding the behavior of functions and their differences. When dealing with the difference of two functions, ${f(x)}$ and ${g(x)}$ , it is essential to determine the intervals where the value of ${(f-g)(x)}$ is negative. This knowledge can be applied in various mathematical contexts, such as optimization problems, graphing functions, and solving inequalities.

Understanding the Difference of Functions

The difference of two functions, ${f(x)}$ and ${g(x)}$ , is defined as ${(f-g)(x) = f(x) - g(x)}$ . This operation involves subtracting the value of ${g(x)}$ from the value of ${f(x)}$ at a given point ${x}$ . The resulting function, ${(f-g)(x)}$ , represents the difference between the two original functions.

Analyzing the Sign of ${(f-g)(x)}$

To determine the intervals where the value of ${(f-g)(x)}$ is negative, we need to analyze the sign of the difference function. The sign of a function at a point ${x}$ can be determined by evaluating the function at that point. If the function value is positive, the sign is positive; if the function value is negative, the sign is negative.

Case 1: ${f(x) > g(x)}$

When ${f(x) > g(x)}$ , the difference function ${(f-g)(x)}$ is positive. This is because we are subtracting a smaller value from a larger value, resulting in a positive difference.

Case 2: ${f(x) < g(x)}$

When ${f(x) < g(x)}$ , the difference function ${(f-g)(x)}$ is negative. This is because we are subtracting a larger value from a smaller value, resulting in a negative difference.

Case 3: ${f(x) = g(x)}$

When ${f(x) = g(x)}$ , the difference function ${(f-g)(x)}$ is zero. This is because we are subtracting an equal value from itself, resulting in a zero difference.

Determining the Interval

To determine the interval where the value of ${(f-g)(x)}$ is negative, we need to find the values of ${x}$ for which ${f(x) < g(x)}$ . This can be done by graphing the functions ${f(x)}$ and ${g(x)}$ and identifying the intervals where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ .

Example

Suppose we have two functions, ${f(x) = 2x + 1}$ and ${g(x) = x^2 - 2}$ . To determine the interval where the value of ${(f-g)(x)}$ is negative, we need to find the values of ${x}$ for which ${f(x) < g(x)}$ .

Step 1: Graph the Functions

Graph the functions ${f(x) = 2x + 1}$ and ${g(x) = x^2 - 2}$ to visualize their behavior.

Step 2: Identify the Intervals

Identify the intervals where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ . This can be done by finding the points of intersection between the two graphs and determining the intervals where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ .

Step 3: Determine the Interval

Determine the interval where the value of ${(f-g)(x)}$ is negative. This can be done by identifying the interval where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ .

Solution

The graph of ${f(x) = 2x + 1}$ lies below the graph of ${g(x) = x^2 - 2}$ in the interval ${(2, ∞)}$ . Therefore, the value of ${(f-g)(x)}$ is negative in the interval ${(2, ∞)}$ .

Conclusion

In conclusion, the value of ${(f-g)(x)}$ is negative in the interval where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ . This can be determined by graphing the functions and identifying the intervals where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ . The correct answer is D. ${(2, ∞)}$ .

Final Answer

The final answer is D. ${(2, ∞)}$ .

Introduction

In our previous article, we discussed the concept of intervals and how to determine the interval where the value of ${(f-g)(x)}$ is negative. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on this topic.

Q1: What is the difference of two functions?

A1: The difference of two functions, ${f(x)}$ and ${g(x)}$ , is defined as ${(f-g)(x) = f(x) - g(x)}$ . This operation involves subtracting the value of ${g(x)}$ from the value of ${f(x)}$ at a given point ${x}$ .

Q2: How do I determine the interval where the value of ${(f-g)(x)}$ is negative?

A2: To determine the interval where the value of ${(f-g)(x)}$ is negative, you need to graph the functions ${f(x)}$ and ${g(x)}$ and identify the intervals where the graph of ${f(x)}$ lies below the graph of ${g(x)}$ .

Q3: What if the functions ${f(x)}$ and ${g(x)}$ intersect at a point?

A3: If the functions ${f(x)}$ and ${g(x)}$ intersect at a point, the value of ${(f-g)(x)}$ will be zero at that point. However, this does not affect the interval where the value of ${(f-g)(x)}$ is negative.

Q4: Can I use algebraic methods to determine the interval where the value of ${(f-g)(x)}$ is negative?

A4: Yes, you can use algebraic methods to determine the interval where the value of ${(f-g)(x)}$ is negative. For example, you can solve the inequality ${f(x) < g(x)}$ to find the values of ${x}$ for which the value of ${(f-g)(x)}$ is negative.

Q5: What if the functions ${f(x)}$ and ${g(x)}$ are not continuous?

A5: If the functions ${f(x)}$ and ${g(x)}$ are not continuous, you may need to use different methods to determine the interval where the value of ${(f-g)(x)}$ is negative. For example, you can use the concept of limits to determine the behavior of the function at a point.

Q6: Can I use technology to graph the functions and determine the interval where the value of ${(f-g)(x)}$ is negative?

A6: Yes, you can use technology such as graphing calculators or computer software to graph the functions and determine the interval where the value of ${(f-g)(x)}$ is negative.

Q7: What if I am given a system of inequalities involving the functions ${f(x)}$ and ${g(x)}$ ?

A7: If you are given a system of inequalities involving the functions ${f(x)}$ and ${g(x)}$ , you can use algebraic methods or graphical methods to determine the interval where the value of ${(f-g)(x)}$ is negative.

Q8: Can I use the concept of intervals to solve optimization problems?

A8: Yes, you can use the concept of intervals to solve optimization problems. For example, you can use the concept of intervals to find the maximum or minimum value of a function.

Q9: What if I am given a function that is not in the form of ${f(x) = ax + b}$ ? Can I still use the concept of intervals?

A9: Yes, you can still use the concept of intervals even if the function is not in the form of ${f(x) = ax + b}$ . You can use algebraic methods or graphical methods to determine the interval where the value of ${(f-g)(x)}$ is negative.

Q10: Can I use the concept of intervals to solve problems involving multiple functions?

A10: Yes, you can use the concept of intervals to solve problems involving multiple functions. For example, you can use the concept of intervals to find the intersection points of multiple functions.

Conclusion

In conclusion, the concept of intervals is a powerful tool for determining the behavior of functions and their differences. By using algebraic methods, graphical methods, or technology, you can determine the interval where the value of ${(f-g)(x)}$ is negative. We hope this Q&A article has provided additional information and clarification on this topic.