Question

determine the values for x when the given function is decreasing
options:
200 ≤ x ≤ 300
10 ≤ x ≤ 12
5 ≤ x ≤ 6
-200 ≤ x ≤ 400
5 ≤ x ≤ 10

Explanation:

Step1: Understand decreasing function

A function is decreasing when as \( x \) increases, \( y \) decreases. We analyze the graph's slope (direction) over intervals.

Step2: Analyze each option

• Option 200 ≤ x ≤ 300: These are \( y \)-values, not \( x \)-values. Eliminate.
• Option 10 ≤ x ≤ 12: From \( x = 10 \) to \( x = 12 \), the graph first decreases (to \( x = 11 \)) then increases (from \( x = 11 \) to \( x = 12 \))? Wait, no—wait, the graph at \( x = 7 \) to \( x = 10 \): wait, let's re-examine. Wait, the key is to look at \( x \)-intervals where \( y \) decreases as \( x \) increases.
• Option 5 ≤ x ≤ 6: From \( x = 5 \) to \( x = 6 \), does \( y \) decrease? Let's check the graph: at \( x = 5 \), \( y \approx 300 \); at \( x = 6 \), \( y \approx 200 \)? Wait, no—wait the graph: from \( x = 5 \) to \( x = 6 \) (or the point at \( x = 6 \)? Wait, the points: \( x = 5 \) (y=300), \( x = 6 \) (y=200? Wait, no, the graph has a point at \( x = 5 \), then \( x = 6 \) (maybe), then \( x = 7 \) (y=400? Wait, no, the graph: let's list the key points. The graph starts at (0, -300), goes up to (3, 100), (5, 300), (6, 200), (7, 400), (9, 0), (10, -200), (11, 0), (12, 100)? Wait, maybe my initial reading was wrong. Wait, the correct way: a function is decreasing when the graph is going downward (from left to right). So intervals where the slope is negative.

Looking at the options:

• 5 ≤ x ≤ 6: Let's see, from \( x = 5 \) to \( x = 6 \), does the graph decrease? If at \( x = 5 \), \( y \) is higher than at \( x = 6 \), then yes. Let's check the options again. The other options: 10 ≤ x ≤ 12: from \( x = 10 \) to \( x = 12 \), the graph goes from (10, -200) to (11, 0) to (12, 100)? Wait, no—wait the graph at \( x = 9 \) is 0, \( x = 10 \) is -200 (decreasing from 9 to 10), then \( x = 11 \) is 0 (increasing from 10 to 11), then \( x = 12 \) is 100 (increasing from 11 to 12). So 10 ≤ x ≤ 12 has both decreasing and increasing. 5 ≤ x ≤ 6: from \( x = 5 \) (y=300) to \( x = 6 \) (y=200), that's decreasing. Wait, but also check 5 ≤ x ≤ 10? Wait, no, from \( x = 7 \) to \( x = 10 \), the graph goes from (7, 400) to (9, 0) to (10, -200), which is decreasing. But the option 5 ≤ x ≤ 10: from \( x = 5 \) to \( x = 7 \), the graph goes from (5, 300) to (6, 200) (decreasing) then (6, 200) to (7, 400) (increasing). So 5 ≤ x ≤ 10 has both decreasing and increasing. Wait, maybe I misread the graph. Let's re-express:

The key is to find the interval where as \( x \) increases, \( y \) decreases. Let's check each option:

1. 200 ≤ x ≤ 300: These are y-values, not x. Eliminate.
2. 10 ≤ x ≤ 12: From x=10 to x=11, y goes from -200 to 0 (increasing). From x=11 to x=12, y goes from 0 to 100 (increasing). So not decreasing. Eliminate.
3. 5 ≤ x ≤ 6: From x=5 (y=300) to x=6 (y=200), y decreases as x increases. This is a decreasing interval.
4. -200 ≤ x ≤ 400: These are y-values, not x. Eliminate.
5. 5 ≤ x ≤ 10: From x=5 to x=7, y goes from 300 to 200 (decreasing) then 200 to 400 (increasing) then 400 to 0 (decreasing) then 0 to -200 (decreasing). So mixed, not entirely decreasing.

So the correct interval is 5 ≤ x ≤ 6.

Answer:

\( 5 \leq x \leq 6 \) (corresponding to the option "5 ≤ x ≤ 6")