analyzing the intervals of a function predict which statements are true about the intervals of the…

analyzing the intervals of a function predict which statements are true about the intervals of the continuous function. check all that apply. x f(x) -3 -15 -2 0 -1 3 0 0 1 -3 2 0 3 15 f(x) > 0 over the interval (-∞, 3). f(x) ≤ 0 over the interval 0, 2. f(x) < 0 over the interval (-1, 1). f(x) > 0 over the interval (-2, 0). f(x) ≥ 0 over the interval 2, ∞).

analyzing the intervals of a function predict which statements are true about the intervals of the continuous function. check all that apply. x f(x) -3 -15 -2 0 -1 3 0 0 1 -3 2 0 3 15 f(x) > 0 over the interval (-∞, 3). f(x) ≤ 0 over the interval 0, 2. f(x) < 0 over the interval (-1, 1). f(x) > 0 over the interval (-2, 0). f(x) ≥ 0 over the interval 2, ∞).

Answer

Explanation:

Step1: Analyze $f(x)>0$ over $(-\infty,3)$

When $x = - 3,f(-3)=-15<0$, so $f(x)>0$ over $(-\infty,3)$ is false.

Step2: Analyze $f(x)\leq0$ over $[0,2]$

When $x = 0,f(0)=0$; when $x = 1,f(1)=-3<0$; when $x = 2,f(2)=0$. So $f(x)\leq0$ over $[0,2]$ is true.

Step3: Analyze $f(x)<0$ over $(-1,1)$

When $x=-1,f(-1) = 3>0$, so $f(x)<0$ over $(-1,1)$ is false.

Step4: Analyze $f(x)>0$ over $(-2,0)$

When $x=-1,f(-1)=3>0$ and when $x = 0,f(0)=0$, so $f(x)>0$ over $(-2,0)$ is true for $x\in(-2,0)$ excluding $x = 0$.

Step5: Analyze $f(x)\geq0$ over $[2,\infty)$

When $x = 2,f(2)=0$ and when $x = 3,f(3)=15>0$. Since the function is continuous, $f(x)\geq0$ over $[2,\infty)$ is true.

Answer:

$f(x)\leq0$ over the interval $[0,2]$, $f(x)>0$ over the interval $(-2,0)$, $f(x)\geq0$ over the interval $[2,\infty)$