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

Explanation:

Step1: Recall decreasing - function definition

A function $y = f(x)$ is decreasing on an interval if for any two values $x_1$ and $x_2$ in the interval with $x_1<x_2$, we have $f(x_1)>f(x_2)$.

Step2: Check the first table

For $x$ values in the interval $(-1,1)$ (i.e., $x = 0$), when $x=-1,f(-1) = 3$ and when $x = 0,f(0)=0$ and when $x = 1,f(1)=-3$. Also, check outside the interval: for $x=-2,f(-2)=0$ and for $x = 2,f(2)=0$. The function is decreasing on $(-1,1)$ and not decreasing elsewhere in the given $x$ - values.

Step3: Check the second table

For $x\in(-1,1)$: when $x=-1,f(-1)=8$ and when $x = 0,f(0)=0$ and when $x = 1,f(1)=-8$. But for $x=-2,f(-2)=10$ and $f(-2)>f(-1)$, so it is decreasing before $x=-1$ too.

Step4: Check the third table

For $x\in(-1,1)$: when $x=-1,f(-1)=-3$ and when $x = 0,f(0)=0$ and when $x = 1,f(1)=3$. The function is increasing on $(-1,1)$.

Step5: Check the fourth table

For $x\in(-1,1)$: when $x=-1,f(-1)=-8$ and when $x = 0,f(0)=0$ and when $x = 1,f(1)=8$. The function is increasing on $(-1,1)$.

Answer:

The first table (with $x$ values - 2, - 1, 0, 1, 2 and $f(x)$ values 0, 3, 0, - 3, 0)