Which table shows a function that is decreasing only over the interval $(-1,1)$?
Table 1:
| x | f(x) |
| --- | ----- |
| -2 | 0 |
| -1 | 3 |
| 0 | 0 |
| 1 | -3 |
| 2 | 0 |
Table 2:
| x | f(x) |
| --- | ----- |
| -2 | 10 |
| -1 | 8 |
| 0 | 0 |
| 1 | -8 |
| 2 | -10 |
Table 3:
| x | f(x) |
| --- | ----- |
| -2 | 0 |
| -1 | -3 |
| 0 | 0 |
| 1 | 3 |
| 2 | 0 |
Table 4:
| x | f(x) |
| --- | ----- |
| -2 | -10 |
| -1 | -8 |
| 0 | 0 |
| 1 | 8 |
| 2 | 10 |
Answer (2)
Analyze the first table and determine that the function is decreasing over the interval ( − 1 , 1 ) .
Check the first table outside the interval ( − 1 , 1 ) and determine that the function is increasing.
Analyze the other tables and determine that they are either increasing over the interval ( − 1 , 1 ) or decreasing outside the interval ( − 1 , 1 ) .
Conclude that the first table shows a function that is decreasing only over the interval ( − 1 , 1 ) .
The answer is the first table.
Explanation
Understanding the Problem We are given four tables of x and f ( x ) values. We want to find the table that represents a function that is decreasing only over the interval ( − 1 , 1 ) . A function is decreasing over an interval if, for any x 1 and x 2 in the interval with x 1 < x 2 , we have f(x_2)"> f ( x 1 ) > f ( x 2 ) .
Analyzing the First Table Let's analyze the first table:
x
f(x)
-2
0
-1
3
0
0
1
-3
2
0
Over the interval ( − 1 , 1 ) , we have f ( − 1 ) = 3 , f ( 0 ) = 0 , and f ( 1 ) = − 3 . Since 0 > -3"> 3 > 0 > − 3 , the function is decreasing from x = − 1 to x = 0 and from x = 0 to x = 1 . Thus, it is decreasing over ( − 1 , 1 ) .
Now let's check outside the interval ( − 1 , 1 ) . We have f ( − 2 ) = 0 and f ( − 1 ) = 3 . Since 0 < 3 , the function is increasing from x = − 2 to x = − 1 . We have f ( 1 ) = − 3 and f ( 2 ) = 0 . Since − 3 < 0 , the function is increasing from x = 1 to x = 2 . Therefore, the function is decreasing only over the interval ( − 1 , 1 ) .
Analyzing the Second Table Let's analyze the second table:
x
f(x)
-2
10
-1
8
0
0
1
-8
2
-10
Over the interval ( − 1 , 1 ) , we have f ( − 1 ) = 8 , f ( 0 ) = 0 , and f ( 1 ) = − 8 . Since 0 > -8"> 8 > 0 > − 8 , the function is decreasing from x = − 1 to x = 0 and from x = 0 to x = 1 . Thus, it is decreasing over ( − 1 , 1 ) .
Now let's check outside the interval ( − 1 , 1 ) . We have f ( − 2 ) = 10 and f ( − 1 ) = 8 . Since 8"> 10 > 8 , the function is decreasing from x = − 2 to x = − 1 . We have f ( 1 ) = − 8 and f ( 2 ) = − 10 . Since -10"> − 8 > − 10 , the function is decreasing from x = 1 to x = 2 . Therefore, the function is decreasing outside the interval ( − 1 , 1 ) .
Analyzing the Third Table Let's analyze the third table:
x
f(x)
-2
0
-1
-3
0
0
1
3
2
0
Over the interval ( − 1 , 1 ) , we have f ( − 1 ) = − 3 , f ( 0 ) = 0 , and f ( 1 ) = 3 . Since − 3 < 0 < 3 , the function is increasing from x = − 1 to x = 0 and from x = 0 to x = 1 . Thus, it is increasing over ( − 1 , 1 ) .
Analyzing the Fourth Table Let's analyze the fourth table:
x
f(x)
-2
-10
-1
-8
0
0
1
8
2
10
Over the interval ( − 1 , 1 ) , we have f ( − 1 ) = − 8 , f ( 0 ) = 0 , and f ( 1 ) = 8 . Since − 8 < 0 < 8 , the function is increasing from x = − 1 to x = 0 and from x = 0 to x = 1 . Thus, it is increasing over ( − 1 , 1 ) .
Conclusion The first table shows a function that is decreasing only over the interval ( − 1 , 1 ) .
Examples
Understanding where a function increases or decreases is crucial in many real-world applications. For example, in economics, it helps analyze the growth or decline of a company's revenue over time. In physics, it can describe the change in velocity of an object. By identifying intervals of increase and decrease, we can make informed decisions and predictions in various fields.
The first table shows a function that is decreasing only over the interval ( − 1 , 1 ) . The values confirm the function decreases from f ( − 1 ) = 3 to f ( 1 ) = − 3 , while increasing outside this interval. Therefore, the answer is Table 1.
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