Which Table Shows A Function That Is Decreasing Only Over The Interval ( − 1 , 1 (-1,1 ( − 1 , 1 ]?1. \[ \begin{tabular}{|c|c|} \hline X$ & F ( X ) F(x) F ( X ) \ \hline -2 & 0 \ -1 & 3 \ 0 & 0 \ 1 & -3 \ 2 & 0
Which Table Shows a Function That is Decreasing Only Over the Interval ?
In mathematics, a function is considered decreasing over a specific interval if its value decreases as the input increases within that interval. In this article, we will examine a table of functions and determine which one shows a function that is decreasing only over the interval .
Understanding Decreasing Functions
A decreasing function is a function that takes on lower values as the input increases. In other words, as the input increases, the output decreases. This is in contrast to an increasing function, where the output increases as the input increases.
Analyzing the Tables
We are given five tables of functions, each with a different function defined. We need to analyze each table and determine if the function is decreasing over the interval .
Table 1
-2 | 0 |
-1 | 3 |
0 | 0 |
1 | -3 |
2 | 0 |
Is the function in Table 1 decreasing over the interval ?
To determine if the function is decreasing, we need to examine the values of the function at different points within the interval. Let's start by examining the values at and . We see that and . Since , the function is not decreasing at this point.
However, we also need to examine the values at and . We see that and . Since , the function is not decreasing at this point.
Therefore, the function in Table 1 is not decreasing over the interval .
Table 2
-2 | 0 |
-1 | 3 |
0 | 0 |
1 | -3 |
2 | 0 |
Is the function in Table 2 decreasing over the interval ?
Let's examine the values of the function at different points within the interval. We see that and . Since , the function is not decreasing at this point.
However, we also need to examine the values at and . We see that and . Since , the function is not decreasing at this point.
Therefore, the function in Table 2 is not decreasing over the interval .
Table 3
-2 | 0 |
-1 | 3 |
0 | 0 |
1 | -3 |
2 | 0 |
Is the function in Table 3 decreasing over the interval ?
Let's examine the values of the function at different points within the interval. We see that and . Since , the function is not decreasing at this point.
However, we also need to examine the values at and . We see that and . Since , the function is not decreasing at this point.
Therefore, the function in Table 3 is not decreasing over the interval .
Table 4
-2 | 0 |
-1 | 3 |
0 | 0 |
1 | -3 |
2 | 0 |
Is the function in Table 4 decreasing over the interval ?
Let's examine the values of the function at different points within the interval. We see that and . Since , the function is not decreasing at this point.
However, we also need to examine the values at and . We see that and . Since , the function is not decreasing at this point.
Therefore, the function in Table 4 is not decreasing over the interval .
Table 5
-2 | 0 |
-1 | 3 |
0 | 0 |
1 | -3 |
2 | 0 |
Is the function in Table 5 decreasing over the interval ?
Let's examine the values of the function at different points within the interval. We see that and . Since , the function is not decreasing at this point.
However, we also need to examine the values at and . We see that and . Since , the function is not decreasing at this point.
Therefore, the function in Table 5 is not decreasing over the interval .
After analyzing all five tables, we can conclude that none of the functions in the tables are decreasing over the interval . However, we can see that the function in Table 1 has a local maximum at and a local minimum at . This suggests that the function may be decreasing over a smaller interval, such as or .
Therefore, the answer to the question is that none of the tables show a function that is decreasing only over the interval .
Q&A: Decreasing Functions and Intervals
In our previous article, we analyzed five tables of functions and determined that none of the functions in the tables are decreasing over the interval . However, we did find that the function in Table 1 has a local maximum at and a local minimum at . This suggests that the function may be decreasing over a smaller interval, such as or .
In this article, we will answer some common questions about decreasing functions and intervals.
Q: What is a decreasing function?
A: A decreasing function is a function that takes on lower values as the input increases. In other words, as the input increases, the output decreases.
Q: How do I determine if a function is decreasing over a specific interval?
A: To determine if a function is decreasing over a specific interval, you need to examine the values of the function at different points within the interval. If the function takes on lower values as the input increases, then it is decreasing over that interval.
Q: What is a local maximum or minimum?
A: A local maximum or minimum is a point on the graph of a function where the function takes on its highest or lowest value within a small interval. For example, if a function has a local maximum at , then the function takes on its highest value at within a small interval around .
Q: How do I find the local maximum or minimum of a function?
A: To find the local maximum or minimum of a function, you need to examine the values of the function at different points within the interval. You can use calculus to find the critical points of the function, which are the points where the function takes on its highest or lowest value.
Q: What is the difference between a local maximum and a global maximum?
A: A local maximum is a point on the graph of a function where the function takes on its highest value within a small interval. A global maximum is a point on the graph of a function where the function takes on its highest value over its entire domain.
Q: How do I determine if a function is decreasing over a specific interval using calculus?
A: To determine if a function is decreasing over a specific interval using calculus, you need to examine the derivative of the function. If the derivative is negative over the interval, then the function is decreasing over that interval.
Q: What is the derivative of a function?
A: The derivative of a function is a measure of how fast the function changes as the input increases. It is denoted by the symbol and is calculated using the limit definition of a derivative.
Q: How do I calculate the derivative of a function?
A: To calculate the derivative of a function, you need to use the limit definition of a derivative. This involves taking the limit of the difference quotient as the change in the input approaches zero.
In this article, we have answered some common questions about decreasing functions and intervals. We have discussed what a decreasing function is, how to determine if a function is decreasing over a specific interval, and how to find the local maximum or minimum of a function. We have also discussed the difference between a local maximum and a global maximum, and how to determine if a function is decreasing over a specific interval using calculus.
We hope that this article has been helpful in answering your questions about decreasing functions and intervals. If you have any further questions, please don't hesitate to ask.