which table shows a function that is decreasing only over the interval (-1, ∞)?

Answer:

We need to check each table to see if the function is decreasing only over the interval ((- 1,\infty)). For a function (y = f(x)) to be decreasing on an interval, if (x_1<x_2) in the interval, then (f(x_1)>f(x_2)).

Let's analyze each table one - by - one:

Table 1:

(x)	(f(x))
(2)	(2)
(1)	(1)
(0)	(0)
(-1)	(- 2)
(-2)	(-3)
(-3)	(x)
The function is decreasing for (x> - 1) (since (f(2)>f(1)>f(0)>f(-1))), but it is also decreasing for (x < - 1) ((f(-1)>f(-2))), so this is not the correct table.

Table 2:

(x)	(f(x))
(2)	(-1)
(1)	(-2)
(0)	(-5)
(-1)	(-1)
(-2)	(-3)
The function is decreasing for (x\in(0,1)) and (x\in(1,2)), but (f(-2)<f(-1)), and (f(-1)<f(0)), so it is not decreasing only over ((-1,\infty))

Table 3:

(x)	(f(x))
(2)	(-8)
(1)	(-4)
(0)	(0)
(-1)	(1)
(-2)	(-1)
(-3)	(-5)
For (x < - 1), (f(-3)<f(-2)) and (f(-2)<f(-1)). For (x>-1), (f(0)>f(1)>f(2)). The function is decreasing only for (x > - 1)

Table 4:

(x)	(f(x))
(2)	(-6)
(1)	(1)
(0)	(2)
(-1)	(-1)
(-2)	(-3)
(-3)	(-4)
The function is not decreasing for (x\in(0,1)) since (f(0)<f(1))

So the table that shows a function that is decreasing only over the interval ((-1,\infty)) is the third table.

Explanation:

Step1: Recall decreasing - function definition

A function (y = f(x)) is decreasing on an interval ((a,b)) if for any (x_1,x_2\in(a,b)) with (x_1 < x_2), (f(x_1)>f(x_2))

Step2: Analyze first table

Check values for (x > - 1) and (x < - 1). It is decreasing for (x < - 1) too, so it's not correct.

Step3: Analyze second table

There are non - decreasing parts for (x < - 1) and inconsistent behavior, so it's not correct.

Step4: Analyze third table

For (x < - 1), the function is non - decreasing and for (x>-1) it is decreasing, so it meets the criteria.

Step5: Analyze fourth table

It has an increasing part for (x\in(0,1)), so it's not correct.