Question

x | f(x)
-3 | -15
-2 | -5
-1 | 0
0 | 5
1 | 0
2 | -5
which is a valid prediction about the continuous function f(x)?
○ f(x) ≤ 0 over the interval (-∞, ∞).
○ f(x) > 0 over the interval (-1, ∞).
○ f(x) ≥ 0 over the interval -1, 1.
○ f(x) < 0 over the interval (0, 2).

Explanation:

• For the option \( f(x) \leq 0 \) over \( (-\infty, \infty) \): From the table, \( f(0) = 5>0 \), so this is false.
• For the option \( f(x)>0 \) over \( (-1, \infty) \): At \( x = 2 \), \( f(2)=- 5<0 \), so this is false.
• For the option \( f(x)\geq0 \) over \( [-1,1] \): At \( x=-1 \), \( f(-1) = 0 \); at \( x = 0 \), \( f(0)=5>0 \); at \( x = 1 \), \( f(1)=0 \). All values in this interval are non - negative, so this is true.
• For the option \( f(x)<0 \) over \( (0,2) \): At \( x = 0 \), \( f(0) = 5>0 \) (and at \( x=1 \), \( f(1) = 0 \)), so this is false.

Answer:

\( f(x) \geq 0 \) over the interval \([-1, 1]\)