Question

over what interval is the function increasing, and over what interval is the function decreasing?

x	f(x)=0.3x²	(x,y)
-2	1.2	(-2,1.2)
-1	0.3	(-1,0.3)
0	0	(0,0)
1	0.3	(1,0.3)
2	1.2	(2,1.2)

the function f(x) is increasing over the interval x≥0.
(simplify your answer. type an inequality.)

Explanation:

Step1: Analyze the function's behavior from the table

Looking at the values of \( x \) and \( f(x) = 0.3x^2 \):

• When \( x \) increases from \(-2\) to \( 0 \) (i.e., \( x \) is negative or zero), \( f(x) \) decreases from \( 1.2 \) to \( 0 \).
• When \( x \) increases from \( 0 \) to \( 2 \) (i.e., \( x \) is zero or positive), \( f(x) \) increases from \( 0 \) to \( 1.2 \).

Step2: Determine the decreasing interval

For the function to be decreasing, as \( x \) increases, \( f(x) \) decreases. From the table, when \( x \) goes from \(-2\) to \( 0 \) ( \( x \) values: \(-2, -1, 0\) ), \( f(x) \) values: \( 1.2, 0.3, 0 \) which is a decreasing trend. So the function is decreasing when \( x \leq 0 \).

Step3: Confirm the increasing interval (already partially given)

The function is increasing when \( x \geq 0 \) as seen from \( x = 0 \) to \( x = 2 \), \( f(x) \) increases from \( 0 \) to \( 1.2 \).

Answer:

The function \( f(x) \) is increasing over the interval \( x \geq 0 \) and decreasing over the interval \( x \leq 0 \).