Decreasing Functions and Intervals

Oct 13, 2025 by ludoplayers7.com 106 views

Understanding the Basics of Intervals and Functions

When analyzing functions, it's essential to understand the concept of intervals and how they relate to the behavior of the function. An interval is a set of real numbers that includes all numbers between two given numbers, along with the endpoints. In the context of functions, intervals are used to describe the domain and range of the function, as well as the behavior of the function within those intervals.

Decreasing Functions and Intervals

A decreasing function is a function that takes on smaller and smaller values as the input increases. In other words, as the input variable x increases, the output value f(x) decreases. To determine if a function is decreasing over a particular interval, we need to examine the behavior of the function within that interval.

Analyzing the Table of Values

Let's examine the table of values below:

Table 1

x	f(x)
-2	0
-1	-5
0	0
1	5

Which Table Shows a Function that is Decreasing over the Interval (-2,0)?

To determine which table shows a function that is decreasing over the interval (-2,0), we need to examine the behavior of the function within that interval.

In Table 1, we can see that as x increases from -2 to 0, f(x) decreases from 0 to 0. However, this is not a decreasing function, as the output value remains the same.
In Table 2, we can see that as x increases from -2 to 0, f(x) decreases from 0 to -5. This is a decreasing function, as the output value decreases as the input variable increases.

Table 2

x	f(x)
-2	0
-1	-5
0	-5
1	5

Conclusion

Based on the analysis above, we can conclude that Table 2 shows a function that is decreasing over the interval (-2,0).

Key Takeaways

Intervals are used to describe the domain and range of a function, as well as the behavior of the function within those intervals.
A decreasing function is a function that takes on smaller and smaller values as the input increases.
To determine if a function is decreasing over a particular interval, we need to examine the behavior of the function within that interval.

Real-World Applications

Understanding intervals and functions is crucial in various real-world applications, such as:

Physics: Intervals are used to describe the motion of objects, including the position, velocity, and acceleration of the object.
Engineering: Intervals are used to describe the behavior of complex systems, including the performance of electronic circuits and the behavior of mechanical systems.
Economics: Intervals are used to describe the behavior of economic systems, including the supply and demand of goods and services.

Conclusion

In conclusion, understanding intervals and functions is essential in various fields, including mathematics, physics, engineering, and economics. By analyzing the behavior of functions within intervals, we can gain valuable insights into the behavior of complex systems and make informed decisions.

References

Calculus: A First Course, by Michael Spivak
Mathematics for Engineers and Scientists, by Donald R. Hill
Economics: Principles, Problems, and Policies, by Gregory Mankiw

Q&A: Intervals and Functions

Frequently Asked Questions

Q: What is an interval in mathematics?

A: An interval is a set of real numbers that includes all numbers between two given numbers, along with the endpoints.

Q: What is a decreasing function?

A: A decreasing function is a function that takes on smaller and smaller values as the input increases.

Q: How do I determine if a function is decreasing over a particular interval?

A: To determine if a function is decreasing over a particular interval, you need to examine the behavior of the function within that interval.

Q: What is the difference between a decreasing function and a non-decreasing function?

A: A decreasing function takes on smaller and smaller values as the input increases, while a non-decreasing function takes on values that are either decreasing or constant.

Q: Can a function be both increasing and decreasing over the same interval?

A: No, a function cannot be both increasing and decreasing over the same interval.

Q: How do I determine if a function is increasing or decreasing over a particular interval?

A: To determine if a function is increasing or decreasing over a particular interval, you need to examine the behavior of the function within that interval.

Q: What is the significance of intervals in mathematics?

A: Intervals are used to describe the domain and range of a function, as well as the behavior of the function within those intervals.

Q: Can intervals be used to describe the behavior of complex systems?

A: Yes, intervals can be used to describe the behavior of complex systems, including the performance of electronic circuits and the behavior of mechanical systems.

Q: How do I apply intervals to real-world problems?

A: To apply intervals to real-world problems, you need to identify the relevant variables and parameters, and then use intervals to describe the behavior of the system.

Q: What are some common applications of intervals in mathematics?

A: Some common applications of intervals in mathematics include physics, engineering, and economics.

Q: Can intervals be used to describe the behavior of economic systems?

A: Yes, intervals can be used to describe the behavior of economic systems, including the supply and demand of goods and services.

Q: How do I use intervals to make informed decisions in real-world problems?

A: To use intervals to make informed decisions in real-world problems, you need to identify the relevant variables and parameters, and then use intervals to describe the behavior of the system.

Conclusion

In conclusion, intervals and functions are essential concepts in mathematics that have numerous applications in various fields. By understanding intervals and functions, you can gain valuable insights into the behavior of complex systems and make informed decisions.

References

Calculus: A First Course, by Michael Spivak
Mathematics for Engineers and Scientists, by Donald R. Hill
Economics: Principles, Problems, and Policies, by Gregory Mankiw

Practice Problems

To practice working with intervals and functions, try the following problems:

Determine if the function f(x) = 2x - 3 is increasing or decreasing over the interval [0, 5].
Find the interval over which the function f(x) = x^2 - 4 is decreasing.
Determine if the function f(x) = 3x + 2 is increasing or decreasing over the interval [-2, 3].

Answer Key

The function f(x) = 2x - 3 is increasing over the interval [0, 5].
The function f(x) = x^2 - 4 is decreasing over the interval [-2, 1].
The function f(x) = 3x + 2 is increasing over the interval [-2, 3].