analyzing an interval\nwhich table shows a function that is decreasing only over the interval (-1, 1)?

Explanation:

Step1: Recall the definition of a decreasing function

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), we have (f(x_1)>f(x_2)).

Step2: Analyze the first table

For (x=-2) to (x = - 1), (f(-2)=0<f(-1) = 3) (increasing). For (x=-1) to (x=0), (f(-1)=3>f(0)=0) (decreasing). For (x = 0) to (x=1), (f(0)=0>f(1)=-3) (decreasing). So it is decreasing on ((-1,2)) not only on ((-1,1)).

Step3: Analyze the second table

For (x=-2) to (x=-1), (f(-2) = 10>f(-1)=8) (decreasing). For (x=-1) to (x = 0), (f(-1)=8>f(0)=0) (decreasing). For (x=0) to (x=1), (f(0)=0>f(1)=-8) (decreasing). So it is decreasing on ((-2,1)) not only on ((-1,1)).

Step4: Analyze the third table

For (x=-2) to (x=-1), (f(-2)=0>f(-1)=-3) (decreasing). For (x=-1) to (x = 0), (f(-1)=-3<f(0)=0) (increasing). For (x=0) to (x=1), (f(0)=0<f(1)=3) (increasing). So it is not decreasing on ((-1,1)).

Step5: Analyze the fourth table

For (x=-2) to (x=-1), (f(-2)=-10<f(-1)=-8) (increasing). For (x=-1) to (x = 0), (f(-1)=-8>f(0)=0) (decreasing). For (x=0) to (x=1), (f(0)=0>f(1)=8) (increasing). So it is decreasing only on the interval ((-1,1)).

Answer:

The fourth table.